Pearson Edexcel IAL Mathematics P4.7.5 distance between two points

Practise finding distances between points from coordinate differences, including simplified surds and 3D vector-length constraints.

Exam points

calculate AB from coordinate differences and give the distance as a simplified surd
set up a squared length equation to find possible parameters or positions on a line

[Maximum number: 3]

The line $l_1$ has equation
$\mathbf r=\begin{pmatrix}2\\3\\-7\end{pmatrix}+\lambda\begin{pmatrix}1\\2\\2\end{pmatrix}$
where $\lambda$ is a scalar parameter.

The line $l_2$ has equation
$\mathbf r=\begin{pmatrix}2\\3\\-7\end{pmatrix}+\mu\begin{pmatrix}4\\-1\\8\end{pmatrix}$
where $\mu$ is a scalar parameter.

Given that $l_1$ and $l_2$ meet at the point P.

The point Q lies on $l_1$ where $\lambda=6$ .
Given that point R lies on $l_2$ such that triangle QPR is an isosceles triangle with PQ=PR.

find the coordinates of the possible positions of point R

Attempts a correct method of finding at least one value for $\mu$ , for example
$(4\mu)^2+\mu^2+(8\mu)^2=6^2+12^2+12^2, \quad \mu=\pm2$
M1
Attempts one correct position:
$\mathbf r=\begin{pmatrix}2\\3\\-7\end{pmatrix}\pm2\begin{pmatrix}4\\-1\\8\end{pmatrix}$
dM1
Possible coordinates are (10,1,9) and (-6,5,-23).
A1

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P4.7.5 - distance between two points question 1