Area identities and a cross-product degeneracy · source reference

Question

Relative to O, A and B have position vectors a,b, nonzero and nonparallel.

(i) State the shortest distance from B to the line through O,A. [1]

(ii) Given $\mathbf c=\mathbf a+\mathbf b$, state the geometrical meaning of $|\mathbf b\times\mathbf c|$ and show $|\mathbf b\times\mathbf c|=|\mathbf a\times\mathbf b|$. [2]

(iii) Given $\mathbf a\times2\mathbf b=\mathbf d\times3\mathbf a$, find a linear relationship between a,b,d. [2]

[2018/DHS/Prelim/II/1]

Source working

(i) Shortest distance $=|\mathbf b\times\hat{\mathbf a}|$ or $|\mathbf b\times\mathbf a|/|\mathbf a|$.

(ii) $|\mathbf b\times\mathbf c|$ is the area of a parallelogram with sides OB and OC.

$$\begin{aligned}|\mathbf b\times\mathbf c|&=|\mathbf b\times(\mathbf a+\mathbf b)|\\&=|(\mathbf b\times\mathbf a)+(\mathbf b\times\mathbf b)|\\&=|\mathbf b\times\mathbf a|=|-(\mathbf a\times\mathbf b)|=|\mathbf a\times\mathbf b|.\end{aligned}$$

The parallelograms with sides OB,OC and OA,OB have the same base and vertical height.

(iii)

$$\mathbf a\times2\mathbf b=\mathbf d\times3\mathbf a,$$ $$(\mathbf a\times2\mathbf b)-(\mathbf d\times3\mathbf a)=\mathbf0,$$ $$\mathbf a\times(2\mathbf b+3\mathbf d)=\mathbf0.$$

Printed conclusion: a is parallel to $2\mathbf b+3\mathbf d$, so $\mathbf a=k(2\mathbf b+3\mathbf d)$, $k\in\mathbb R$.

Teacher notes

The shortest distance is perpendicular BN from B to OA.

For parallelogram OBDC,

$$|\mathbf b\times\mathbf c|=|\mathbf b||\mathbf c|\sin\theta=(OB)(h)=2\times\text{Area}(OBC)=\text{Area}(OBDC).$$

For (iii), the margin writes

$$\mathbf a\times(2\mathbf b+3\mathbf d)=(|\mathbf a||2\mathbf b+3\mathbf d|\sin\theta)\hat{\mathbf n}=\mathbf0,$$

where θ is between a and $2\mathbf b+3\mathbf d$, and $\hat{\mathbf n}$ is perpendicular to both. It concludes $\sin\theta=0$, $\theta=0^\circ$ or $180^\circ$, so parallel.

Qualification of printed relationship

The source omits $2\mathbf b+3\mathbf d=\mathbf0$, where that angle is undefined and a cannot equal a scalar times the zero vector. The inclusive relationship is $2\mathbf b+3\mathbf d=t\mathbf a$ for real $t$, allowing $t=0$.

Native diagrams

The distance panel has O,N,A on a line, B above N and right angle at N. The area panel shows A,C,D collinear, O,B on the parallel base, dotted AC extension and AB/CB diagonals, red h and its right angle; a separate margin parallelogram OBDC labels θ between OB,OC and the same height h.

Printed answer key

(i) $|\mathbf b\times\hat{\mathbf a}|$ or $|\mathbf b\times\mathbf a|/|\mathbf a|$; (ii) area of a parallelogram with sides OB and OC; (iii) $\mathbf a=k(2\mathbf b+3\mathbf d)$, $k\in\mathbb R$. See the qualification above for the omitted zero-vector case.

Provenance

2025 TJC H2 Math (9758) JC2 MSM: VECTORS. Question document pages 2; solution document pages 7. Original page numbers are retained as text metadata.