CAIE A-Level Further Math AS 1.6 Vectors Questions

Practise using vector products, plane equations, lines and distances in 3D geometry, with answers in Cartesian and vector forms.

Exam points

find plane normals using vector products, then form ax+by+cz=d from a point
calculate shortest distances between skew lines using a common perpendicular vector
find line or plane intersections by combining normals, points and vector equations

Covered syllabus points

1.6.1—Plane equations 1.6.2—Vector product 1.6.3—Lines and planes

[Maximum number: 9]

The lines $l_{1}$ and $l_{2}$ have equations

\mathbf{r}=-2 \mathbf{i}-3 \mathbf{j}-5 \mathbf{k}+\lambda(-4 \mathbf{i}+3 \mathbf{j}+5 \mathbf{k}) \quad \text { and } \quad \mathbf{r}=2 \mathbf{i}-2 \mathbf{j}+3 \mathbf{k}+\mu(2 \mathbf{i}-3 \mathbf{j}+\mathbf{k})

respectively.

(a)

Find the shortest distance between $l_{1}$ and $l_{2}$ .

The plane $\Pi$ contains $l_{1}$ and the point with position vector -i-3 j-4 k.

[ 5 ]

$\left(\begin{array}{c}2 -2 3\end{array}\right)-\left(\begin{array}{l}-2 -3 -5\end{array}\right)=\left(\begin{array}{l}4 1 8\end{array}\right)$
Finds direction of one line to another.

$\left|\begin{array}{ccc}\mathbf{i} \mathbf{j} \mathbf{k} -4 3 2 -3 1\end{array}\right|=\left(\begin{array}{c}18 14 6\end{array}\right) \sim\left(\begin{array}{l}9 7 3\end{array}\right)$
Find common perpendicular.

$\frac{1}{\sqrt{139}}\left|\left(\begin{array}{l}4 1 8\end{array}\right) \bullet\left(\begin{array}{l}9 7 3\end{array}\right)\right|=\frac{67}{\sqrt{139}}(=5.68)$
Uses formula for shortest distance.

5

(b)

Find an equation of $\Pi$ , giving your answer in the form a x+b y+c z=d.

[ 4 ]

$\left|\begin{array}{ccc}\mathbf{i} \mathbf{j} \mathbf{k} 1 0 -4 3 5\end{array}\right|=\left(\begin{array}{c}3 9 -3\end{array}\right) \sim\left(\begin{array}{c}1 3 -1\end{array}\right)$
Finds vector perpendicular to the plane.

$1(-1)+3(-3)-1(-4)=-6 \Rightarrow x+3 y-z=-6$
Uses point in the plane.

4

[Maximum number: 13]

The plane $\Pi_{1}$ has equation $\mathbf{r}=\mathbf{i}-\mathbf{j}-2 \mathbf{k}+\lambda(\mathbf{i}-2 \mathbf{j}-3 \mathbf{k})+\mu(3 \mathbf{i}-\mathbf{k})$ .

(a)

Find an equation for $\Pi_{1}$ in the form a x+b y+c z=d.

The line l, which does not lie in $\Pi_{1}$ , has equation r=-3 i+k+t(i+j+k).

[ 4 ]

$\left|\begin{array}{ccc}\mathbf{i} \mathbf{j} \mathbf{k} 1 -2 -3 3 0 -1\end{array}\right|=\left(\begin{array}{c}2 -8 6\end{array}\right) \sim\left(\begin{array}{c}1 -4 3\end{array}\right)$
Finds perpendicular to $\Pi_{1}$ .

1(1)-4(-1)+3(-2)=-1
Uses point on $\Pi_{1}$ .

x-4 y+3 z=-1
4

(b)

Show that l is parallel to $\Pi_{1}$ .

[ 2 ]

$\left(\begin{array}{c}1 -4 3\end{array}\right)\left(\begin{array}{l}1 1 1\end{array}\right)=1-4+3=0$
Shows dot product with direction of
line is 0.

2

(c)

Find the distance between l and $\Pi_{1}$ .

[ 3 ]

$\frac{1}{\sqrt{1^{2}+4^{2}+3^{2}}}\left(\begin{array}{c}-4 1 3\end{array}\right) \cdot\left(\begin{array}{c}1 -4 3\end{array}\right)$ or $\frac{1}{\sqrt{1^{2}+4^{2}+3^{2}}}\left(\left(\begin{array}{c}-3 0 1\end{array}\right) \cdot\left(\begin{array}{c}1 -4 3\end{array}\right)+1\right)$
Uses correct formula for distance from point on l to $\Pi_{1}$ .
$\frac{1}{\sqrt{1^{2}+4^{2}+3^{2}}}(-3.1+0.-4+1.3+1)$ $\frac{1}{\sqrt{26}}(=0.196)$
3

(d)

The plane $\Pi_{2}$ has equation 3 x+3 y+2 z=1.

Find a vector equation of the line of intersection of $\Pi_{1}$ and $\Pi_{2}$ .

[ 4 ]

States point common to both planes e.g. $\left(\begin{array}{c}\frac{1}{15} \frac{4}{15} 0\end{array}\right)$ .
$\left(\begin{array}{c}\frac{5}{7} 0 \frac{-4}{7}\end{array}\right)$ or $\left(\begin{array}{c}0 \frac{5}{17} \frac{1}{17}\end{array}\right)$ or alternative.

$\left|\begin{array}{ccc}\mathbf{i} \mathbf{j} \mathbf{k} 1 -4 3 3 2\end{array}\right|=\left(\begin{array}{c}-17 7 15\end{array}\right)$
Finds direction of line.

$\mathbf{r}=\left(\begin{array}{c}\frac{5}{7} 0 -\frac{4}{7}\end{array}\right)+\lambda\left(\begin{array}{c}-17 7 15\end{array}\right)$
OE.

4

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Covered syllabus points

Question 1

Question (a)

Question (b)

Question 2

Question (a)

Question (b)

Question (c)

Question (d)