Relative to a fixed origin $O$ ,

a straight laser beam path $l$ passes through station $A$ with position vector $i−2j+5k\mathbf{i} - 2\mathbf{j} + 5\mathbf{k}$
the line $l$ also passes through station $B$ with position vector $4i−4j+2k4\mathbf{i} - 4\mathbf{j} + 2\mathbf{k}$
a diagnostic probe is located at point $P$ with position vector $−4i−6j+10k-4\mathbf{i} - 6\mathbf{j} + 10\mathbf{k}$

Find a vector equation for $l$ .

The point $C$ lies on $l$ such that $P C$ is perpendicular to $l$ .

Find the coordinates of $C$ .

A secondary probe $P^{'}$ is positioned such that $P^{'}$ is the reflection of $P$ in the line $l$ .

Find the coordinates of $P^{'}$ .

Hence find $∣PP′⃗∣|\vec{PP'}|$ , giving your answer as a simplified surd.

Relative to a fixed origin $O$ ,

a straight laser beam path $l$ passes through station $A$ with position vector $i−2j+5k\mathbf{i} - 2\mathbf{j} + 5\mathbf{k}$
the line $l$ also passes through station $B$ with position vector $4i−4j+2k4\mathbf{i} - 4\mathbf{j} + 2\mathbf{k}$
a diagnostic probe is located at point $P$ with position vector $−4i−6j+10k-4\mathbf{i} - 6\mathbf{j} + 10\mathbf{k}$

Find a vector equation for $l$ .

The point $C$ lies on $l$ such that $P C$ is perpendicular to $l$ .

Find the coordinates of $C$ .

A secondary probe $P^{'}$ is positioned such that $P^{'}$ is the reflection of $P$ in the line $l$ .

Find the coordinates of $P^{'}$ .

Hence find $∣PP′⃗∣|\vec{PP'}|$ , giving your answer as a simplified surd.

4.11 Vectors (A-level only)

4.11 Vectors (A-level only) Questions