Relative to the origin $O$, the points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b

Question

Relative to the origin $O$, the points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$ respectively, where $\mathbf{a}$ and $\mathbf{b}$ are non-zero vectors. It is given that $\mathbf{a}	imes \mathbf{b}=\mathbf{0}$. Find $\mathbf{a}\cdot \mathbf{b}$ in terms of $\left| \mathbf{b} ight|$. [2] It is now given that $\mathbf{a}	imes \mathbf{b}
e \mathbf{0}$. Point $P$ is the foot of the perpendicular from $A$ to the line $OB$ and the point $Q$ is the foot of the perpendicular from $B$ to the line $OA$. It is given that $AP=BQ$. Write down the lengths $AP$ and $BQ$ in terms of $\mathbf{a}$ and $\mathbf{b}$. [2] Hence show that $\left| \mathbf{a} ight|=\left| \mathbf{b} ight|$. [1] Given also that the angle between $\mathbf{a}$ and $\mathbf{a}-\mathbf{b}$ is $\phi $ radians, show that $\mathbf{a}\cdot \mathbf{b}=-{{\left| \mathbf{a} ight|}^{2}}\cos \,2\phi $. [2] The pyramid $ORST$ has a triangular base $ORS$ and height $OT$. The position vectors of $R$ and $S$ are $-4\mathbf{i}+\mathbf{j}+3\mathbf{k}$ and $5\mathbf{i}+\mathbf{j}$ respectively. Find the possible coordinates of $T$ if the volume of the pyramid $ORST$ is $35$ units$^{3}$. [5] [Volume of pyramid $=\frac{1}{3}	imes $base area$	imes $height]

Accepted Answer

(a) n = (-3, -15, -9); (b) T = (-2, -10, -6) or (2, 10, 6)

2020 TJC P2 Q4

2020 TJC P2 Q4

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