Translate the expression into a length, area, direction, line, plane or locus. Name the actual points and lines in your answer. Let A, B and P have position vectors \(\mathbf a,\mathbf b,\mathbf p\). Direction and normal vectors are non-zero; points defining a line are distinct.
Lengths and position vectors
| Expression or condition | Geometrical interpretation |
|---|---|
| \(\mathbf b-\mathbf a\) | Displacement vector \(\overrightarrow{AB}\). |
| \(|\mathbf b-\mathbf a|\) | Distance AB. |
| \(\dfrac{\mathbf a+\mathbf b}{2}\) | Position vector of the midpoint of AB. |
| \(\dfrac{\mathbf b-\mathbf a}{|\mathbf b-\mathbf a|}\) | Unit vector pointing from A towards B. |
Scalar products and projections
Let \(\widehat{\mathbf b}=\dfrac{\mathbf b}{|\mathbf b|}\). The denominator identifies the direction onto which the other vector is projected.
| Expression or condition | Geometrical interpretation |
|---|---|
| \(\mathbf a\cdot\widehat{\mathbf b}\) | Signed scalar projection of OA in the direction of OB. |
| \(|\mathbf a\cdot\widehat{\mathbf b}|=\dfrac{|\mathbf a\cdot\mathbf b|}{|\mathbf b|}\) | Length of the projection of OA onto line OB. |
| \((\mathbf a\cdot\widehat{\mathbf b})\widehat{\mathbf b}\) | Vector projection of \(\mathbf a\) onto \(\mathbf b\). |
| \(\mathbf a\cdot\mathbf b=0\) | The non-zero vectors are perpendicular. |
| \((\mathbf p-\mathbf a)\cdot(\mathbf b-\mathbf a)=0\) | AP is perpendicular to AB, provided P differs from A. |
Exam check: \(|\mathbf a\cdot\mathbf b|\) is not generally a projection length. If \(|\mathbf a|=1\), it is the projection length of OB onto OA. If \(|\mathbf b|=1\), it is the projection length of OA onto OB. Absolute value removes the sign; a length cannot be negative.
Vector products, areas and perpendicular distances
| Expression or condition | Geometrical interpretation |
|---|---|
| \(\mathbf a\times\mathbf b\) | For non-parallel vectors: a vector perpendicular to both, oriented by the right-hand rule. |
| \(|\mathbf a\times\mathbf b|\) | Area of the parallelogram with adjacent sides OA and OB; twice the area of triangle OAB. |
| \(\dfrac12|(\mathbf b-\mathbf a)\times(\mathbf p-\mathbf a)|\) | Area of triangle ABP. |
| \(|\mathbf a\times\widehat{\mathbf b}|=\dfrac{|\mathbf a\times\mathbf b|}{|\mathbf b|}\) | Perpendicular distance from A to line OB. |
| \(\dfrac{|(\mathbf p-\mathbf a)\times\mathbf d|}{|\mathbf d|}\) | Perpendicular distance from P to the line through A parallel to \(\mathbf d\). |
| \(\mathbf a\times\mathbf b=\mathbf0\) | The non-zero vectors are parallel (same or opposite directions). |
Why the distance formula works: \(|\mathbf a\times\mathbf b|=|\mathbf b|(|\mathbf a|\sin\theta)\) is base times perpendicular height. Divide by the base length to obtain the height. The foot may lie on the extended line, outside the segment.
Recognising line equations
| Expression or condition | Geometrical interpretation |
|---|---|
| \(\mathbf r=\mathbf a+t\mathbf d,\quad t\in\mathbb R\) | Line through A parallel to \(\mathbf d\). |
| \(\mathbf r=(1-t)\mathbf a+t\mathbf b\) | Whole line AB for \(t\in\mathbb R\); segment AB including endpoints for \(0\le t\le1\). |
| \((\mathbf r-\mathbf a)\times\mathbf d=\mathbf0\) | Line through A parallel to \(\mathbf d\). |
| \(\mathbf r\times\mathbf d=\mathbf a\times\mathbf d\) | Rearrange to \((\mathbf r-\mathbf a)\times\mathbf d=\mathbf0\): the same line. |
| \(\mathbf r\times(\mathbf b-\mathbf a)=\mathbf0\) | Line through O parallel to AB; it need not be line AB. |
| \((\mathbf r-\mathbf a)\times(\mathbf r-\mathbf b)=\mathbf0\) | Line AB, including A and B. |
Recognising planes and their distances
| Expression or condition | Geometrical interpretation |
|---|---|
| \((\mathbf r-\mathbf a)\cdot\mathbf n=0\) | Plane through A with normal vector \(\mathbf n\). |
| \(\mathbf r\cdot\mathbf n=d\) | Plane normal to \(\mathbf n\), at distance \(\dfrac{|d|}{|\mathbf n|}\) from O. |
| \(\mathbf r\cdot\widehat{\mathbf n}=d\) | With a unit normal, d is the signed offset; the distance from O is \(|d|\). |
| \(\mathbf r=\mathbf a+s\mathbf u+t\mathbf v\) | Plane through A parallel to non-parallel directions \(\mathbf u,\mathbf v\), for real s and t. |
| \(\dfrac{|\mathbf p\cdot\mathbf n-d|}{|\mathbf n|}\) | Perpendicular distance from P to the plane \(\mathbf r\cdot\mathbf n=d\). |
Interpreting simultaneous solutions
| Expression or condition | Geometrical interpretation |
|---|---|
| Two lines: one common point | Intersecting lines. |
| Two lines: no common points | Distinct parallel lines or skew lines. |
| Two lines: infinitely many common points | Coincident lines. |
| Line and plane: one common point | The line meets the plane at one point. |
| Line and plane: no common points | The line is parallel to and outside the plane. |
| Line and plane: infinitely many common points | The whole line lies in the plane. |
| Two planes | Distinct parallel planes have no common points; otherwise they intersect in a line or coincide. They cannot have exactly one common point. |
Loci and useful deductions
| Expression or condition | Geometrical interpretation |
|---|---|
| \(|\mathbf r-\mathbf a|=|\mathbf r-\mathbf b|\) | In 3D: the perpendicular bisector plane of AB. |
| \(|\mathbf r-\mathbf a|=k,\quad k>0\) | Sphere with centre A and radius k in 3D. In a plane through A: a circle. |
| \((\mathbf r-\mathbf a)\cdot(\mathbf r-\mathbf b)=0\) | Sphere with diameter AB in 3D; in a specified plane containing AB: the circle with diameter AB. |
| \((\mathbf a+\mathbf b)\cdot(\mathbf a-\mathbf b)=0\) | The sum and difference vectors are perpendicular; these are the diagonal directions of the parallelogram with adjacent sides \(\mathbf a,\mathbf b\). |
| \(\dfrac{\mathbf a}{|\mathbf a|}+\dfrac{\mathbf b}{|\mathbf b|}\) | Direction of the internal angle bisector of angle AOB, for non-collinear rays OA and OB. |
Exam check: apply every restriction on the moving point. A sphere intersected by a plane may give a circle, one point or no points. A line equation with a restricted parameter may describe only a segment or ray.
Answering precisely
Identify the quantity or object; check for a unit vector; rewrite position-vector differences as named displacements; name the actual line, plane, triangle or projection; include parameter restrictions. For example, \(|\mathbf b\times\mathbf c|\) is the area of the parallelogram with adjacent sides OB and OC, or twice the area of triangle OBC.
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