Edexcel A-Level Mathematics A2 Fp3 5 2 Use of Vectors in Problems Questions

Practise applying vectors to lines, planes, distances, intersections and angles, choosing forms that make the 3D geometry efficient.

Exam points

substitute a line into a plane to find an intersection point or parameter value
find a line of intersection of two planes using a point and a direction vector
calculate shortest distances or acute angles using normals and direction vectors

[Maximum number: 5]

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

l_{1}: \mathbf{r}=(\mathbf{i}+2 \mathbf{j}-5 \mathbf{k})+\lambda(5 \mathbf{i}+\mathbf{j})

and

l_{2}: \mathbf{r}=(2 \mathbf{i}-4 \mathbf{j}+4 \mathbf{k})+\mu(8 \mathbf{i}-2 \mathbf{j}+3 \mathbf{k})

where $\lambda$ and $\mu$ are scalar parameters.

Determine the shortest distance between $l_{1}$ and $l_{2}$

Give your answer in simplest form.

Way 1:
$\frac{6}{\sqrt{3^2+15^2+18^2}}=\frac{2}{\sqrt{62}}, \quad \frac{21}{\sqrt{62}}$ $d=\frac{2}{\sqrt{62}}+\frac{21}{\sqrt{62}} =\frac{23}{\sqrt{62}} =\frac{23\sqrt{62}}{62}$

OR

Way 2:
$\overrightarrow{AB}=\pm(-\mathbf i+6\mathbf j-9\mathbf k)$ $d=\frac{|\overrightarrow{AB}\cdot\mathbf n|}{|\mathbf n|} =\frac{69}{\sqrt{558}} =\frac{23}{\sqrt{62}}$

All question bank results loaded

Edexcel A-Level Mathematics A2 Fp3 5 2 Use of Vectors in Problems Questions question 1