Practice section 1 of 6

The line l passes through P and Q. The position vector of P is $\mathbf p$ and $\overrightarrow{PQ}=\mathbf m$.

(a) Express the position vector of Q in terms of p,m.

(b) E lies on PQ produced with PE=3 PQ; F is the midpoint of PQ; G lies on QP produced with PG:PQ=2:1. (i) Mark E,F,G on the diagram. (ii) Express their position vectors in terms of p,m.

(c) What can you conclude about the position vector of any point R on l?

Find a vector equation and a Cartesian equation of the line passing through $(3,-4,7)$ and $(5,-4,6)$.

A line has equation $(x-5)/3=1-y=2z+4$. Find a vector equation of the line.

Self-Practice 7:

(a) Convert $\mathbf r=(2,3,-1)+\lambda(3,4,5)$, λ real, to Cartesian form.

(b) Convert $\mathbf r=\mathbf i+\mathbf j+\lambda(\mathbf i+5\mathbf k)$, λ real, to Cartesian form.

(c) Convert $(x-1)/2=y/3=z-4$ to a vector equation.

(d) Convert $(1-x)/3=(2y-1)/5$, z=4, to a vector equation.

(e) Give vector and Cartesian equations for the line through A with position vector $\mathbf a=\mathbf i-3\mathbf j+2\mathbf k$, parallel to $\mathbf m=5\mathbf i+4\mathbf j-\mathbf k$.

Determine whether $(-1,3,-3)$ lies on the line in Example 21: $\mathbf r=(5,1,-2)+\lambda'(6,-2,1)$, $\lambda'\in\mathbb R$, equivalently $(x-5)/3=1-y=2z+4$.

The point is $P=(1,1,-1)$ and the line is $\mathbf r=(1,2,3)+\lambda(1,2,1)$, $\lambda\in\mathbb R$.

(i) Find the coordinates of the foot of the perpendicular from P to the line.

(ii) Deduce the shortest distance from P to the line.

(iii) Find the position vector of the image of P in the line.