With respect to the origin $O$, the position vectors of the points $A$, $B$ and $C$ are $\mathbf{a}$

Question

With respect to the origin $O$, the position vectors of the points $A$, $B$ and $C$ are $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ respectively. Point $C$ lies on $AB$ such that $AC:CB=1:2$. It is given that $\mathbf{a}$ is a unit vector and the length of $OB$ is $2$ units. Give a geometrical interpretation of $\left| \mathbf{a}\cdot \mathbf{c} \right|$. [1] It is given that the angle $AOB$ is ${{60}^{\circ }}$. By considering $\left( 2\mathbf{a}-\mathbf{b} \right)\cdot \left( 2\mathbf{a}-\mathbf{b} \right)$, find $\left| 2\mathbf{a}-\mathbf{b} \right|$. [3] Find $\mathbf{c}$ in terms of $\mathbf{a}$ and $\mathbf{b}$. [1] Hence by considering cosine of angle $AOC$ and cosine of angle $COB$, determine if the line segment $OC$ bisects the angle $AOB$. [3]

Accepted Answer

(i) length of projection of OC to line OA; (ii) 2; (iii) c = (2a+b)/3; (iv) OC bisects angle AOB (shown)

2021 MI P1 Q4

2021 MI P1 Q4

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