Give a geometrical interpretation of what it means when solving the equations \(\mathbf{r}=\mathbf{a

Question

Give a geometrical interpretation of what it means when solving the equations $\mathbf{r}=\mathbf{a}+t\mathbf{b}$ and $\mathbf{r}\cdot \mathbf{n}=k$ gives infinitely many solutions, where $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{n}$ are constant vectors, $t$ is a real parameter and $k$ is a constant scalar.[1] solving the equations $\mathbf{r}=\mathbf{a}+t\mathbf{b}$ and $\mathbf{r}=\mathbf{c}+m\mathbf{d}$ does not give a unique solution, where $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$ and $\mathbf{d}$ are constant vectors, $t$ and $m$ are real parameters.[1] A plane $p$ contains the point $A$ with position vector $3\mathbf{i}+9\mathbf{k}$ and is perpendicular to vector $-4\mathbf{i}+5\mathbf{j}+3\mathbf{k}$. Find the exact shortest distance from the origin $O$ to plane $p$.[2] By considering the plane containing the point $D$ with coordinates $\left( 3,-2,4 \right)$ and parallel to $p$ or otherwise, determine whether the origin $O$ and $D$ are on the same or opposite side of $p$.[2] Show that the coordinates of the foot of the perpendicular from $D$ to $p$ is $\left( 1,\frac{1}{2},\frac{11}{2} \right)$.[2] Hence find an equation that describes the set of all points which are of the same distance from $p$ as $D$ but on the opposite side of $p$.[2]

Accepted Answer

(a)(i) The line lies in the plane. (ii) The lines are parallel (possibly coincident) or skew. (b)(i) $\frac3{\sqrt2}\text{ units}$. (ii) $O$ and $D$ are on the same side of $p$. (iii) $\left(1,\frac12,\frac{11}{2}\right)$. (iv) $-4x+5y+3z=40$.

2020 TMJC P1 Q7

2020 TMJC P1 Q7

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