Edexcel A-Level Mathematics A2 P4.7.2 Magnitude of a Vector Questions

Practise Edexcel IAL P4.7.2 vector magnitude: calculate lengths from components, connect magnitudes to geometry, and form unit vectors.

Exam points

Calculate |a| from components for exact lengths or surds
Link vector magnitudes to distances, areas 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

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

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Edexcel A-Level Mathematics A2 P4.7.2 Magnitude of a Vector Questions question 1