Pearson Edexcel IAL Mathematics P4.7.2 Magnitude of a vector

Practise using vector magnitude to find lengths, unit vectors, distances from points and triangle areas in 2D or 3D.

Exam points

calculate |a| from components and use it for exact distances or simplified surds
link magnitudes to triangle areas, closest-point problems or parameter equations
find a unit vector by dividing a direction vector by its magnitude

[Maximum number: 3]

With respect to a fixed origin O, the line $l_{1}$ is given by the equation

\mathbf{r}=\mathbf{i}+2 \mathbf{j}+5 \mathbf{k}+\lambda(8 \mathbf{i}-\mathbf{j}+4 \mathbf{k})

where $\lambda$ is a scalar parameter.
The point A lies on $l_{1}$
Given that $|\overrightarrow{O A}|=5 \sqrt{10}$

show that at A the parameter $\lambda$ satisfies

81 \lambda^{2}+52 \lambda-220=0

$|\overrightarrow{OA}|^2=(1+8\lambda)^2+(2-\lambda)^2+(5+4\lambda)^2$
M1
$|\overrightarrow{OA}|=5\sqrt{10} \Rightarrow (1+8\lambda)^2+(2-\lambda)^2+(5+4\lambda)^2=250$
M1
$\begin{aligned} 64\lambda^2+16\lambda+1+\lambda^2-4\lambda+4+16\lambda^2+40\lambda+25&=250\\ 81\lambda^2+52\lambda-220&=0 \end{aligned}$
A1*

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P4.7.2 - Magnitude of a vector question 1