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H1 Math

H2 Math

IB AA HL Math

IB AA SL Math

IB AI HL Math

IB AI SL Math

IP Math

N(A) A Math

N(A) E Math

PSLE

PSLE Foundation Mathematics

<p>The lines \(l_1\) and \(l_2\) have equations \[\mathbf{r}=\begin{pmatrix}2\\1\\-1\end{pmatrix}+\lambda\begin{pmatrix}1\\-2\\3\end{pmatrix}\qquad\text{and}\qquad\mathbf{r}=\begin{pmatrix}3\\4\\p\end{pmatrix}+\mu\begin{pmatrix}2\\1\\q\end{pmatrix}\] respectively, where&nbsp;\(p\) and \(q\) are constants, and \(\lambda\) and \(\mu\) are parameters.</p>
<ol class="abc-list"><li data-list-item-id="nyjc-q11-item01" data-list-item-path="nyjc-q11-item01"><div class="list-item-paragraph">Suppose that&nbsp;\(l_1\) intersects \(l_2\).</div><ol class="i-list"><li data-list-item-id="nyjc-q11-item01-sub-item01" data-list-item-path="nyjc-q11-item01/nyjc-q11-item01-sub-item01"><div class="list-item-paragraph">Find the coordinates of the point of intersection and state a linear relationship between \(p\) and \(q\).<span class="inline-marks">[3]</span></div></li><li data-list-item-id="nyjc-q11-item01-sub-item02" data-list-item-path="nyjc-q11-item01/nyjc-q11-item01-sub-item02"><div class="list-item-paragraph">Find the acute angle between \(l_1\) and \(l_2\) when&nbsp;\(q=2\).<span class="inline-marks">[2]</span></div></li></ol></li><li data-list-item-id="nyjc-q11-item02" data-list-item-path="nyjc-q11-item02"><div class="list-item-paragraph">By using vector product, show that a vector perpendicular to both&nbsp;\(l_1\) and \(l_2\) is \(\begin{pmatrix}2q+3\\q-6\\m\end{pmatrix}\), where&nbsp;\(m\) is a constant to be determined.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="nyjc-q11-item03" data-list-item-path="nyjc-q11-item03"><div class="list-item-paragraph">The plane \(\pi\) contains \(l_2\) and is parallel to \(l_1\). Find the shortest distance between \(l_1\) and \(\pi\) in terms of \(p\) and \(q\).<span class="inline-marks">[4]</span></div></li></ol>

The lines \(l_1\) and \(l_2\) have equations \[\mathbf{r}=\begin{pmatrix}2\\1\\-1\end{pmatrix}+\lambda\begin{pmatrix}1\\-2\\3\end{pmatrix}\qquad\text{and}\qquad\mathbf{r}=\begin{pmatrix}3\\4\\p\end{pmatrix}+\mu\begin{pmatrix}2\\1\\q\end{pmatrix}\] respectively, where \(p\) and \(q\) are constants, and \(\lambda\) and \(\mu\) are parameters.

By using vector product, show that a vector perpendicular to both \(l_1\) and \(l_2\) is \(\begin{pmatrix}2q+3\\q-6\\m\end{pmatrix}\), where \(m\) is a constant to be determined.

The plane \(\pi\) contains \(l_2\) and is parallel to \(l_1\). Find the shortest distance between \(l_1\) and \(\pi\) in terms of \(p\) and \(q\).

Distance from a Point to a Plane

Angles Between Two Lines

Intersection of Two Lines

Vector (Cross) Product

Equations of Planes

2026 NYJC P1 Q11

Vectors

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<p>With reference to the origin \(O\), the points \(A\), \(B\) and \(C\) are such that vectors \(\overrightarrow{OA}=\mathbf{a}\), \(\overrightarrow{OB}=\mathbf{b}\) and \(\overrightarrow{OC}=2\mathbf{a}-2\mathbf{b}\).</p>
<ol class="abc-list"><li data-list-item-id="acjc-q08-item01" data-list-item-path="acjc-q08-item01"><div class="list-item-paragraph">Given that&nbsp;\(|\mathbf{a}|=2\), \(|\mathbf{b}|=3\) and the angle between \(\mathbf{a}\) and \(\mathbf{b}\) is \(\frac{\pi}{6}\), find the area of triangle \(OBC\).<span class="inline-marks">[3]</span></div></li><li data-list-item-id="acjc-q08-item02" data-list-item-path="acjc-q08-item02"><div class="list-item-paragraph">The lines \(OA\) and \(BC\) intersect at point \(X\).</div><div class="list-item-paragraph">Show that&nbsp;\(\displaystyle \overrightarrow{OX}=\frac23\mathbf{a}\).<span class="inline-marks">[3]</span></div></li><li data-list-item-id="acjc-q08-item03" data-list-item-path="acjc-q08-item03"><div class="list-item-paragraph">The plane \(p\) has equation \(\mathbf{r}\cdot\mathbf{n}=d\) and contains the point \(A\), where&nbsp;\(\mathbf{n}\) is a unit vector.</div><div class="list-item-paragraph">Find the shortest distance from point \(X\) to plane \(p\), in the form of \(\frac1k|d|\), where&nbsp;\(k\) is a positive integer to be determined.<span class="inline-marks">[3]</span></div></li></ol>

With reference to the origin \(O\), the points \(A\), \(B\) and \(C\) are such that vectors \(\overrightarrow{OA}=\mathbf{a}\), \(\overrightarrow{OB}=\mathbf{b}\) and \(\overrightarrow{OC}=2\mathbf{a}-2\mathbf{b}\).

Given that \(|\mathbf{a}|=2\), \(|\mathbf{b}|=3\) and the angle between \(\mathbf{a}\) and \(\mathbf{b}\) is \(\frac{\pi}{6}\), find the area of triangle \(OBC\).

Areas of Triangles and Parallelograms Using Vectors

2026 ACJC P1 Q8

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<p>The diagram shows a triangle \(ABC\) with vertices \(A(-5,\ 2)\) and \(B(7,\ 8)\). The point \(C\) lies on the perpendicular bisector of \(AB\) and the equation of the line \(BC\) is \(y=3x-13\).</p>
<ol class="abc-list"><li data-list-item-id="cg-q13-a" data-list-item-path="cg-q13-a"><div class="list-item-paragraph">Find the equation of the perpendicular bisector of \(AB\).<span class="inline-marks">[4]</span></div></li><li data-list-item-id="cg-q13-b" data-list-item-path="cg-q13-b"><div class="list-item-paragraph">Find the coordinates of \(C\).<span class="inline-marks">[2]</span></div></li><li data-list-item-id="cg-q13-c" data-list-item-path="cg-q13-c"><div class="list-item-paragraph">Find the area of the triangle \(ABC\).<span class="inline-marks">[2]</span></div></li></ol>

The diagram shows a triangle \(ABC\) with vertices \(A(-5,\ 2)\) and \(B(7,\ 8)\). The point \(C\) lies on the perpendicular bisector of \(AB\) and the equation of the line \(BC\) is \(y=3x-13\).

Find the equation of the perpendicular bisector of \(AB\).

Area of Rectilinear Figures

Parallel and Perpendicular Lines

Equation of a Straight Line

Midpoint of a Line Segment

Perpendicular Bisectors

Perpendicular bisector and the third vertex of a triangle

Coordinate Geometry

<p>The diagram shows a trapezium \(ABCD\) in which \(A\) is \((1,\ 2)\), \(B\) is \((5,\ 10)\) and \(D\) is \((9,\ 3)\). Angle \(ABC\) is \(90^\circ\) and \(AB\) is parallel to \(DC\).</p>
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<ol class="i-list"><li data-list-item-id="cg-q09-i" data-list-item-path="cg-q09-i"><div class="list-item-paragraph">Find the equation of \(CD\).<span class="inline-marks">[2]</span></div></li><li data-list-item-id="cg-q09-ii" data-list-item-path="cg-q09-ii"><div class="list-item-paragraph">Find the equation of \(BC\).<span class="inline-marks">[2]</span></div></li><li data-list-item-id="cg-q09-iii" data-list-item-path="cg-q09-iii"><div class="list-item-paragraph">Hence find the coordinates of \(C\).<span class="inline-marks">[2]</span></div></li><li data-list-item-id="cg-q09-iv" data-list-item-path="cg-q09-iv"><div class="list-item-paragraph">Find the area of the trapezium \(ABCD\).<span class="inline-marks">[3]</span></div></li><li data-list-item-id="cg-q09-v" data-list-item-path="cg-q09-v"><div class="list-item-paragraph">The point \(E\) lies on \(AC\) such that&nbsp;\(AE : EC = 3 : 2\). Find the coordinates of \(E\).<span class="inline-marks">[2]</span></div></li></ol>

The diagram shows a trapezium \(ABCD\) in which \(A\) is \((1,\ 2)\), \(B\) is \((5,\ 10)\) and \(D\) is \((9,\ 3)\). Angle \(ABC\) is \(90^\circ\) and \(AB\) is parallel to \(DC\).

The point \(E\) lies on \(AC\) such that \(AE : EC = 3 : 2\). Find the coordinates of \(E\).

Ratio Theorem and Section Formula

Right-angled trapezium and a point dividing a diagonal

<p>\(ABCD\) is a rhombus in which \(A\) is \((4,\ 6)\) and \(B\) is \((5,\ 14)\). The diagonals \(AC\) and \(BD\) intersect at \(T\), and \(AC\) cuts the \(y\)-axis at \((0,\ 4)\).</p>
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<ol class="i-list"><li data-list-item-id="cg-q07-i" data-list-item-path="cg-q07-i"><div class="list-item-paragraph">Find the equations of \(AC\) and \(BD\).<span class="inline-marks">[4]</span></div></li><li data-list-item-id="cg-q07-ii" data-list-item-path="cg-q07-ii"><div class="list-item-paragraph">Calculate the coordinates of \(T\), \(C\) and \(D\).<span class="inline-marks">[4]</span></div></li><li data-list-item-id="cg-q07-iii" data-list-item-path="cg-q07-iii"><div class="list-item-paragraph">Find the area of the rhombus \(ABCD\).<span class="inline-marks">[3]</span></div></li></ol>

\(ABCD\) is a rhombus in which \(A\) is \((4,\ 6)\) and \(B\) is \((5,\ 14)\). The diagonals \(AC\) and \(BD\) intersect at \(T\), and \(AC\) cuts the \(y\)-axis at \((0,\ 4)\).

Calculate the coordinates of \(T\), \(C\) and \(D\).

Diagonals of a rhombus

<p>The diagram shows a kite \(ABCD\) in which the side \(AB\) is parallel to the \(x\)-axis. The diagonals \(AC\) and \(BD\) intersect at \(T\).</p>
<div class="graph-block" data-graph-config="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" data-width="440" data-height="250" data-proportional style="width:440px;height:250px;max-width:100%;display:flow-root;margin:0 auto;margin-bottom:0px;--graph-width:440px;--graph-aspect:440 / 250;"></div>
<ol class="i-list"><li data-list-item-id="cg-q06-i" data-list-item-path="cg-q06-i"><div class="list-item-paragraph">It is given that&nbsp;\(B\) is \((7,\ 2)\) and the equation of the diagonal \(AC\) is \(2y=x+7\).</div><ol class="abc-list"><li data-list-item-id="cg-q06-i-a" data-list-item-path="cg-q06-i/cg-q06-i-a"><div class="list-item-paragraph">Find the coordinates of \(A\).<span class="inline-marks">[2]</span></div></li><li data-list-item-id="cg-q06-i-b" data-list-item-path="cg-q06-i/cg-q06-i-b"><div class="list-item-paragraph">Find the equation of the diagonal \(BD\).<span class="inline-marks">[3]</span></div></li><li data-list-item-id="cg-q06-i-c" data-list-item-path="cg-q06-i/cg-q06-i-c"><div class="list-item-paragraph">Hence find the coordinates of \(T\) and of \(D\).<span class="inline-marks">[4]</span></div></li></ol></li><li data-list-item-id="cg-q06-ii" data-list-item-path="cg-q06-ii"><div class="list-item-paragraph">Given also that the coordinates of \(C\) are \((11,\ 9)\), find the area of the kite \(ABCD\).<span class="inline-marks">[3]</span></div></li></ol>

The diagram shows a kite \(ABCD\) in which the side \(AB\) is parallel to the \(x\)-axis. The diagonals \(AC\) and \(BD\) intersect at \(T\).

It is given that \(B\) is \((7,\ 2)\) and the equation of the diagonal \(AC\) is \(2y=x+7\).

Given also that the coordinates of \(C\) are \((11,\ 9)\), find the area of the kite \(ABCD\).

Kite with a side parallel to the x-axis

<p>The diagram shows a rectangle \(ABCD\) in which \(A\) and \(C\) lie on the \(x\)-axis. The equation of \(AB\) is \(3y=x+1\) and the equation of \(BD\) is \(4y=3x-12\).</p>
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<p>Find</p>
<ol class="i-list"><li data-list-item-id="cg-q04-i" data-list-item-path="cg-q04-i"><div class="list-item-paragraph">the coordinates of \(B\),<span class="inline-marks">[2]</span></div></li><li data-list-item-id="cg-q04-ii" data-list-item-path="cg-q04-ii"><div class="list-item-paragraph">the coordinates of the midpoint of \(AC\),<span class="inline-marks">[2]</span></div></li><li data-list-item-id="cg-q04-iii" data-list-item-path="cg-q04-iii"><div class="list-item-paragraph">the coordinates of \(A\), \(C\) and \(D\),<span class="inline-marks">[4]</span></div></li><li data-list-item-id="cg-q04-iv" data-list-item-path="cg-q04-iv"><div class="list-item-paragraph">the length of \(BC\),<span class="inline-marks">[2]</span></div></li><li data-list-item-id="cg-q04-v" data-list-item-path="cg-q04-v"><div class="list-item-paragraph">the area of the rectangle \(ABCD\).<span class="inline-marks">[2]</span></div></li></ol>

The diagram shows a rectangle \(ABCD\) in which \(A\) and \(C\) lie on the \(x\)-axis. The equation of \(AB\) is \(3y=x+1\) and the equation of \(BD\) is \(4y=3x-12\).

the coordinates of the midpoint of \(AC\),

the coordinates of \(A\), \(C\) and \(D\),

Length of a Line Segment

Rectangle with a diagonal along the x-axis

<p>The diagram shows a rectangle \(ABCD\) in which \(A\) is \((1,\ 2)\) and \(D\) is \((5,\ 0)\). The diagonal \(AC\) lies on the line \(y=x+1\).</p>
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<p>Find</p>
<ol class="i-list"><li data-list-item-id="cg-q03-i" data-list-item-path="cg-q03-i"><div class="list-item-paragraph">the equation of \(CD\),<span class="inline-marks">[3]</span></div></li><li data-list-item-id="cg-q03-ii" data-list-item-path="cg-q03-ii"><div class="list-item-paragraph">the coordinates of \(C\) and of \(B\),<span class="inline-marks">[4]</span></div></li><li data-list-item-id="cg-q03-iii" data-list-item-path="cg-q03-iii"><div class="list-item-paragraph">the length of \(AB\),<span class="inline-marks">[2]</span></div></li><li data-list-item-id="cg-q03-iv" data-list-item-path="cg-q03-iv"><div class="list-item-paragraph">the area of the rectangle \(ABCD\).<span class="inline-marks">[2]</span></div></li></ol>

The diagram shows a rectangle \(ABCD\) in which \(A\) is \((1,\ 2)\) and \(D\) is \((5,\ 0)\). The diagonal \(AC\) lies on the line \(y=x+1\).

Rectangle from a vertex pair and a diagonal

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<p>The origin \(O\) and the points \(A\), \(B\) and \(C\) lie in the same plane, where&nbsp;\(\overrightarrow{OA} = \mathbf{a}\), \(\overrightarrow{OB} = \mathbf{b}\) and \(\overrightarrow{OC} = \mathbf{c}\). It is given that&nbsp;\(\angle AOC = \angle BOC = \theta\), where&nbsp;\(\theta\) is an acute angle.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Show that&nbsp;\(\mathbf{c} \cdot \mathbf{\hat{a}} = \mathbf{c} \cdot \mathbf{\hat{b}}\) where&nbsp;\(\mathbf{\hat{a}}\) and \(\mathbf{\hat{b}}\) are unit vectors in the directions of vectors \(\mathbf{a}\) and \(\mathbf{b}\) respectively.<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">If&nbsp;\(\mathbf{c}\) can be written as&nbsp;\(m\mathbf{\hat{a}} + n\mathbf{\hat{b}}\), where&nbsp;\(m\) and \(n\) are constants, use the result from <strong>(a)</strong> to show that&nbsp;\(m = n\).<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Write down the equation of the line passing through the points \(A\) and \(B\).<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-paragraph">Given that&nbsp;\(|\mathbf{a}| = 3\), \(|\mathbf{b}| = 2\) and \(m = n\), show that the position vector of the point of intersection of the line passing through \(A\) and \(B\) and the line passing through \(O\) and \(C\) is \(t\left(\mathbf{\hat{a}} + \mathbf{\hat{b}}\right)\), where&nbsp;\(t\) is a constant to be determined.<span class="inline-marks">[3]</span></div></li></ol>

The origin \(O\) and the points \(A\), \(B\) and \(C\) lie in the same plane, where \(\overrightarrow{OA} = \mathbf{a}\), \(\overrightarrow{OB} = \mathbf{b}\) and \(\overrightarrow{OC} = \mathbf{c}\). It is given that \(\angle AOC = \angle BOC = \theta\), where \(\theta\) is an acute angle.

Angle Bisectors Using Vectors

Equations of Lines

Scalar (Dot) Product

2024 ACJC P2 Q4

<p>With reference to the point \(O\) as the origin and the \(x\)-\(y\) plane as a horizontal plane, the pyramid \(OPQRV\) has a parallelogram base \(OPQR\) and height \(OV\). The position vectors of the points \(P\) and \(R\) are \(-3\mathbf{i} + 4\mathbf{j} + 3\mathbf{k}\) and \(5\mathbf{i} - 2\mathbf{j} - \mathbf{k}\) respectively.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Find the coordinates of the point \(S\) that lies on the line \(PR\) such that the distance from \(O\) to \(S\) is a minimum.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Find the cartesian equations of the planes such that the perpendicular distance from each plane to the base \(OPQR\) is \(2\sqrt{86}\) units.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Find the acute angle between \(OV\) and the vertical.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-paragraph">Given that&nbsp;\(\overrightarrow{QV}\) is parallel to the vector \(-5\mathbf{j} + 8\mathbf{k}\), find the position vector of the point \(V\). Hence find the exact volume of the pyramid \(OPQRV\).<span class="inline-marks">[5]</span></div></li></ol>
<p>\(\text{[Volume of a pyramid} = \frac{1}{3} \times \text{base area} \times \text{height]}\)</p>

With reference to the point \(O\) as the origin and the \(x\)-\(y\) plane as a horizontal plane, the pyramid \(OPQRV\) has a parallelogram base \(OPQR\) and height \(OV\). The position vectors of the points \(P\) and \(R\) are \(-3\mathbf{i} + 4\mathbf{j} + 3\mathbf{k}\) and \(5\mathbf{i} - 2\mathbf{j} - \mathbf{k}\) respectively.

\(\text{[Volume of a pyramid} = \frac{1}{3} \times \text{base area} \times \text{height]}\)

Perpendicular Projection and Reflection in a Line

Three-Dimensional Geometry

Vectors in a Parallelogram

2025 VJC P2 Q3

<p>An airfield on a horizontal flat piece of ground must manage the take-off flight paths of aircraft from its runways. These flight paths may be assumed to be straight lines, and the aircraft may be assumed to be point objects.</p>
<p>In the diagram below, the labelled points are defined relative to an origin \(O\) with coordinates \((0,0,0)\) on the airfield. The \(x\)-, \(y\)- and \(z\)-axes are in the directions east, north and vertically upwards respectively, with units in metres. The direction vectors \(\mathbf{i}\), \(\mathbf{j}\) and \(\mathbf{k}\) correspond to the east, north and vertically upwards directions respectively. The point \(A\) has coordinates \((2000,500,0)\) and the point \(B\) has coordinates \((1400,2900,0)\).</p>
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<p>Aircraft Alpha takes off from Point \(A\) in the direction \(4\mathbf{i} + 3\mathbf{j} + \mathbf{k}\) as shown.</p>
<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">Find the initial take-off angle \(\theta\), the angle that Aircraft Alpha's flight path makes with the horizontal ground.<span class="inline-marks">[2]</span></div></li></ol>
<p>The peak of a majestic mountain has coordinates \((18000,4500,2000)\).</p>
<ol class="i-list" start="2"><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">Find the coordinates of the closest point on the flight path of Aircraft Alpha to the peak. Hence find the closest distance between the flight path and the peak, giving your answer to the nearest metre.<span class="inline-marks">[5]</span></div></li><li data-list-item-id="item-iii" data-list-item-path="item-iii"><div class="list-item-paragraph">Aircraft Bravo takes off from Point \(B\). For aviation safety reasons, the flight paths of Aircraft Alpha and Aircraft Bravo should never cross each other. Given that Aircraft Bravo will take off in the direction \(4\mathbf{i} + 2\mathbf{j} + h\mathbf{k}\), find the value of \(h\) which will result in the flight paths crossing each other, giving your answer to \(3\) decimal places.<span class="inline-marks">[2]</span></div></li></ol>
<p>The value of \(h\) is now given to be \(1\).</p>
<ol class="i-list" start="4"><li data-list-item-id="item-iv" data-list-item-path="item-iv"><div class="list-item-paragraph">Find the cartesian equation of the plane containing point \(B\) and the flight path of Aircraft Alpha. Using this answer, provide another mathematical explanation as to why the flight path of Aircraft Bravo will not cross the flight path of Aircraft Alpha.<span class="inline-marks">[4]</span></div></li></ol>

An airfield on a horizontal flat piece of ground must manage the take-off flight paths of aircraft from its runways. These flight paths may be assumed to be straight lines, and the aircraft may be assumed to be point objects.

In the diagram below, the labelled points are defined relative to an origin \(O\) with coordinates \((0,0,0)\) on the airfield. The \(x\)-, \(y\)- and \(z\)-axes are in the directions east, north and vertically upwards respectively, with units in metres. The direction vectors \(\mathbf{i}\), \(\mathbf{j}\) and \(\mathbf{k}\) correspond to the east, north and vertically upwards directions respectively. The point \(A\) has coordinates \((2000,500,0)\) and the point \(B\) has coordinates \((1400,2900,0)\).

Aircraft Alpha takes off from Point \(A\) in the direction \(4\mathbf{i} + 3\mathbf{j} + \mathbf{k}\) as shown.

The peak of a majestic mountain has coordinates \((18000,4500,2000)\).

The value of \(h\) is now given to be \(1\).

Angle Between a Line and a Plane

Relationships Between a Line and a Plane

Distance from a Point to a Line

2023 ACJC P1 Q12

<p>The line \(l_1\) has equation \(x - 1 = 2 - y = -z\).</p>
<p>The line \(l_2\) passes through the point \(A(-2, 1, 7)\) and is parallel to the vector \(-3\mathbf{i} + \mathbf{j} + 5\mathbf{k}\).</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Write down vector equations for&nbsp;\(l_1\) and \(l_2\).</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Show that the lines \(l_1\) and \(l_2\) intersect and find the coordinates of \(E\), the point of intersection of \(l_1\) and \(l_2\).</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">The acute angle between the lines \(l_1\) and \(l_2\) is denoted by \(\theta\). By finding \(\cos \theta\), show that \[\sin \theta = \frac{2\sqrt{2}}{\sqrt{35}}.\] Hence find the shortest distance from \(A\) to the line \(l_1\), leaving your answer in exact form.</span><span class="inline-marks">[5]</span></div></li></ol>

The line \(l_1\) has equation \(x - 1 = 2 - y = -z\).

The line \(l_2\) passes through the point \(A(-2, 1, 7)\) and is parallel to the vector \(-3\mathbf{i} + \mathbf{j} + 5\mathbf{k}\).

Write down vector equations for \(l_1\) and \(l_2\).

Show that the lines \(l_1\) and \(l_2\) intersect and find the coordinates of \(E\), the point of intersection of \(l_1\) and \(l_2\).

The acute angle between the lines \(l_1\) and \(l_2\) is denoted by \(\theta\). By finding \(\cos \theta\), show that \[\sin \theta = \frac{2\sqrt{2}}{\sqrt{35}}.\] Hence find the shortest distance from \(A\) to the line \(l_1\), leaving your answer in exact form.

<p>The plane \(p\) passes through the points with coordinates \((4, -1, -3)\), \((-2, -5, 2)\) and \((4, -3, -2)\).</p>
<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">Find a Cartesian equation of \(p\).<span class="inline-marks">[4]</span></div></li></ol>
<p>The line \(l_1\) has equation \(\frac{x-1}{2} = \frac{y-2}{-4} = \frac{z+3}{1}\) and the line \(l_2\) has equation \(\frac{x+2}{1} = \frac{y-1}{5} = \frac{z-3}{k}\), where&nbsp;\(k\) is a constant. It is given that&nbsp;\(l_1\) and \(l_2\) intersect.</p>
<ol class="i-list" start="2"><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">Find the value of \(k\).<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-iii" data-list-item-path="item-iii"><div class="list-item-paragraph">Show that&nbsp;\(l_1\) lies in \(p\) and find the coordinates of the point at which \(l_2\) intersects \(p\).<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-iv" data-list-item-path="item-iv"><div class="list-item-paragraph">Find the acute angle between \(l_2\) and \(p\).<span class="inline-marks">[3]</span></div></li></ol>

The plane \(p\) passes through the points with coordinates \((4, -1, -3)\), \((-2, -5, 2)\) and \((4, -3, -2)\).

The line \(l_1\) has equation \(\frac{x-1}{2} = \frac{y-2}{-4} = \frac{z+3}{1}\) and the line \(l_2\) has equation \(\frac{x+2}{1} = \frac{y-1}{5} = \frac{z-3}{k}\), where \(k\) is a constant. It is given that \(l_1\) and \(l_2\) intersect.

Show that \(l_1\) lies in \(p\) and find the coordinates of the point at which \(l_2\) intersects \(p\).

Find the acute angle between \(l_2\) and \(p\).

Intersection of a Line and a Plane

N2011 P1 Q11

<p>The lines \(l_1\) and \(l_2\) have equations</p>
<p style="text-align: center">\[\mathbf{r} = \begin{pmatrix} 0 \\ 1 \\ 2 \end{pmatrix} + s \begin{pmatrix} 1 \\ 0 \\ 3 \end{pmatrix} \text{ and}\hspace{0.5em} \mathbf{r} = \begin{pmatrix} -2 \\ 3 \\ 1 \end{pmatrix} + t \begin{pmatrix} 2 \\ b \\ 0 \end{pmatrix}\]</p>
<p>respectively, where&nbsp;\(b\) is a constant.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">For the case where the lines intersect, calculate the acute angle between them.</span></div></li><li><div class="list-item-content"><span class="list-item-text">For the case when&nbsp;\(b = 1\), find the position vectors of the points \(P\) on \(l_1\) and \(Q\) on \(l_2\) such that&nbsp;\(O\), \(P\) and \(Q\) are collinear, where&nbsp;\(O\) is the origin.</span></div></li></ol>

The lines \(l_1\) and \(l_2\) have equations

\[\mathbf{r} = \begin{pmatrix} 0 \\ 1 \\ 2 \end{pmatrix} + s \begin{pmatrix} 1 \\ 0 \\ 3 \end{pmatrix} \text{ and } \mathbf{r} = \begin{pmatrix} -2 \\ 3 \\ 1 \end{pmatrix} + t \begin{pmatrix} 2 \\ b \\ 0 \end{pmatrix}\]

For the case where the lines intersect, calculate the acute angle between them.

For the case when \(b = 1\), find the position vectors of the points \(P\) on \(l_1\) and \(Q\) on \(l_2\) such that \(O\), \(P\) and \(Q\) are collinear, where \(O\) is the origin.

Collinearity Using Vectors

N2005 P1 Q14

<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1775962953671-l5kobtsrm5n.jpg?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzU5NjI5NTM2NzEtbDVrb2J0c3JtNW4uanBnIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjY5NTgwMywiZXhwIjoxNzkyNjk2MTAzfQ.B0px06EKtVRIsvD82QLJhbVHY4c3TuhGv6wEqoZ9dp0" style="max-width: min(270px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>In the diagram, \(O\) is the centre of the rectangular base \(ABCD\) of a right pyramid with vertex \(V\). Perpendicular unit vectors \(\mathbf{i}\), \(\mathbf{j}\), \(\mathbf{k}\) are parallel to \(AB\), \(BC\) and \(OV\) respectively. The length of \(AB\), \(BC\) and \(OV\) are \(12\text{ cm}\), \(6\text{ cm}\) and \(6\text{ cm}\) respectively. A line \(l\) has Cartesian equation \(\frac{-x-4}{10} = y+2 = \frac{t-z}{-2}.\)</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find a vector equation of \(AV\).</span></div></li><li><div class="list-item-content"><span class="list-item-text">If the line \(l\) intersects \(AV\) at \(M\), find the position vector of \(M\) and the value of \(t\).</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the acute angle between \(AV\) and the line \(l\). Hence find the perpendicular distance from \(A\) to the line \(l\).</span></div></li></ol>

In the diagram, \(O\) is the centre of the rectangular base \(ABCD\) of a right pyramid with vertex \(V\). Perpendicular unit vectors \(\mathbf{i}\), \(\mathbf{j}\), \(\mathbf{k}\) are parallel to \(AB\), \(BC\) and \(OV\) respectively. The length of \(AB\), \(BC\) and \(OV\) are \(12\text{ cm}\), \(6\text{ cm}\) and \(6\text{ cm}\) respectively. A line \(l\) has Cartesian equation \(\frac{-x-4}{10} = y+2 = \frac{t-z}{-2}.\)

If the line \(l\) intersects \(AV\) at \(M\), find the position vector of \(M\) and the value of \(t\).

Find the acute angle between \(AV\) and the line \(l\). Hence find the perpendicular distance from \(A\) to the line \(l\).

2006 HCI P1 Q14

<p>Referred to the origin \(O\), points \(A\) and \(B\) have position vectors \(\mathbf{a}\) and \(\mathbf{b}\) respectively. Point \(C\) lies on \(OA\), between \(O\) and \(A\), such that&nbsp;\(OC : CA = 3 : 2\). Point \(D\) lies on \(OB\), between \(O\) and \(B\), such that&nbsp;\(OD : DB = 5 : 6\).</p>
<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">Find the position vectors \(\overrightarrow{OC}\) and \(\overrightarrow{OD}\), giving your answers in terms of \(\mathbf{a}\) and \(\mathbf{b}\).<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">Show that the vector equation of the line \(BC\) can be written as&nbsp;\(\mathbf{r} = \frac{3}{5} \lambda \mathbf{a} + (1 - \lambda) \mathbf{b}\), where&nbsp;\(\lambda\) is a parameter. Find in similar form the vector equation of the line \(AD\) in terms of a parameter \(\mu\).<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-iii" data-list-item-path="item-iii"><div class="list-item-paragraph">Find, in terms of \(\mathbf{a}\) and \(\mathbf{b}\), the position vector of the point \(E\) where the lines \(BC\) and \(AD\) meet and find the ratio \(AE : ED\).<span class="inline-marks">[5]</span></div></li></ol>

Referred to the origin \(O\), points \(A\) and \(B\) have position vectors \(\mathbf{a}\) and \(\mathbf{b}\) respectively. Point \(C\) lies on \(OA\), between \(O\) and \(A\), such that \(OC : CA = 3 : 2\). Point \(D\) lies on \(OB\), between \(O\) and \(B\), such that \(OD : DB = 5 : 6\).

Show that the vector equation of the line \(BC\) can be written as \(\mathbf{r} = \frac{3}{5} \lambda \mathbf{a} + (1 - \lambda) \mathbf{b}\), where \(\lambda\) is a parameter. Find in similar form the vector equation of the line \(AD\) in terms of a parameter \(\mu\).

Find, in terms of \(\mathbf{a}\) and \(\mathbf{b}\), the position vector of the point \(E\) where the lines \(BC\) and \(AD\) meet and find the ratio \(AE : ED\).

N2015 P1 Q7

<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1782643154194-60z17dgtnt8.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3ODI2NDMxNTQxOTQtNjB6MTdkZ3RudDgucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjY5NTgwMywiZXhwIjoxNzkyNjk2MTAzfQ.ETrfb5PO6aeYvF2qFoZyIfmAByE9V4rFApSOYpkHCbU" style="max-width: min(250px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>The diagram shows a triangle \(PQR\) with vertices \(P(9,10)\) and \(Q(1,-2)\). The point \(R\) lies on the perpendicular bisector of \(PQ\) and the equation of the line \(QR\) is \(y = 8x - 10\).</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-content"><span class="list-item-text">Find the equation of the perpendicular bisector of \(PQ\).</span><span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-content"><span class="list-item-text">Find the coordinates of \(R\).</span><span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-content"><span class="list-item-text">Find the area of triangle \(PQR\).</span><span class="inline-marks">[2]</span></div></li></ol>

The diagram shows a triangle \(PQR\) with vertices \(P(9,10)\) and \(Q(1,-2)\). The point \(R\) lies on the perpendicular bisector of \(PQ\) and the equation of the line \(QR\) is \(y=8x-10\).

Find the equation of the perpendicular bisector of \(PQ\).

2025 DSS PRELIMS P1 Q12

<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1770805603072-58nwvw5dd8k.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NzA4MDU2MDMwNzItNThud3Z3NWRkOGsucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjY5NTgwMywiZXhwIjoxNzkyNjk2MTAzfQ.QnDyKCaVB2TZUdwET4yTRVbuTWMGi2wQb3VCixEUphM" style="max-width: min(250px, 100%); height: auto; display: block;" alt="The diagram shows an isosceles triangle \(PQR\) on a coordinate plane." class="aligncenter custom-width" /></figure></div>
<p>The diagram shows an isosceles triangle \(PQR\) with vertices at \(P(0, k)\), \(Q(3, 9)\) and \(R(h, 6)\), where&nbsp;\(h\) and \(k\) are positive constants and \(h &gt; k\).</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Show that&nbsp;\(h + k = 12\).<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">It is now given that&nbsp;\(h = 7\).<br>The point \(S\) is such that&nbsp;\(PQRS\) is a kite. Given that&nbsp;\(S\) lies on the line \(21y = 7x - 8\), find the coordinates of \(S\).<span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">Find the area of the kite.<span class="inline-marks">[2]</span></div></li></ol>

The diagram shows an isosceles triangle \(PQR\) with vertices at \(P(0, k)\), \(Q(3, 9)\) and \(R(h, 6)\), where \(h\) and \(k\) are positive constants and \(h > k\).

It is now given that \(h = 7\).\nThe point \(S\) is such that \(PQRS\) is a kite. Given that \(S\) lies on the line \(21y = 7x - 8\), find the coordinates of \(S\).

IP3 Class Test (May 2025) Q11 (2023 ACSBR AM Prelim P2 Q5)

<p>The planes \(p_1\) and \(p_2\) have equations \(\mathbf{r} \cdot \begin{pmatrix} 1 \\ 1 \\ -1 \end{pmatrix} = 0\) and \(\mathbf{r} \cdot \begin{pmatrix} 2 \\ -1 \\ -2 \end{pmatrix} = 6\) respectively.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find a vector equation of \(l\), the line of intersection between \(p_1\) and \(p_2\).</span><span class="inline-marks">[2]</span></div></li></ol>
<p>Another line, \(l_1\), has equation \(\mathbf{r} = \begin{pmatrix} 2 \\ 0 \\ 8 \end{pmatrix} + \mu \begin{pmatrix} 1 \\ 1 \\ 5 \end{pmatrix}\), where&nbsp;\(\mu\) is a real parameter.</p>
<ol class="i-list" start="2"><li><div class="list-item-content"><span class="list-item-text">Use the result in part (i) to show that&nbsp;\(p_1\), \(p_2\) and \(l_1\) have exactly one common point of intersection. (You are not allowed to use any calculator for this part of the question.)</span><span class="inline-marks">[3]</span></div></li></ol>

The planes \(p_1\) and \(p_2\) have equations \(\mathbf{r} \cdot \begin{pmatrix} 1 \\ 1 \\ -1 \end{pmatrix} = 0\) and \(\mathbf{r} \cdot \begin{pmatrix} 2 \\ -1 \\ -2 \end{pmatrix} = 6\) respectively.

Find a vector equation of \(l\), the line of intersection between \(p_1\) and \(p_2\).

Another line, \(l_1\), has equation \(\mathbf{r} = \begin{pmatrix} 2 \\ 0 \\ 8 \end{pmatrix} + \mu \begin{pmatrix} 1 \\ 1 \\ 5 \end{pmatrix}\), where \(\mu\) is a real parameter.

Use the result in part (i) to show that \(p_1\), \(p_2\) and \(l_1\) have exactly one common point of intersection. (You are not allowed to use any calculator for this part of the question.)

Line of Intersection of Two Planes

2015 NJC P2 Q3

<p>A circle with centre \(C\) passes through points \(A(-8, 3)\) and \(B(-1, -4)\).</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Explain why the perpendicular bisector of \(AB\) will pass through \(C\).</span></div></li><li><div class="list-item-content"><span class="list-item-text">Given that&nbsp;\(C\) lies on the line \(y = 2x + 5\), find \(C\).</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence find the equation of the circle.</span></div></li></ol>

A circle with centre \(C\) passes through points \(A(-8, 3)\) and \(B(-1, -4)\).

Explain why the perpendicular bisector of \(AB\) will pass through \(C\).

Given that \(C\) lies on the line \(y = 2x + 5\), find \(C\).

Math Question Bank