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

Question

Relative to the origin $O$, the position vectors of points $A$, $B$ and $C$ are $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ respectively. It is given that $\mathbf{a}$ is a unit vector and $\mathbf{b}$ is perpendicular to $\mathbf{b} - \mathbf{a}$. Show that $0 < |\mathbf{b}| < 1$.[3] It is given further that $|\mathbf{b}| = \frac{1}{2}$ and $C$ lies on $AB$ such that $AC : CB = 2 : 1$. Find the value of $|\mathbf{a} \cdot \mathbf{c}|$ and state the geometrical interpretation of $|\mathbf{a} \cdot \mathbf{c}|$.[4] Find the value of $\frac{|\mathbf{a} \times \mathbf{b}|}{|\mathbf{a} \times \mathbf{c}|}$.[2]

Accepted Answer

(a) $0<|\mathbf b|<1$ (b) $|\mathbf a\mathbin{\cdot}\mathbf c|=\dfrac12$, the length of the projection of $\overrightarrow{OC}$ onto $OA$ (c) $\dfrac32$

2025 CJC P1 Q6

2025 CJC P1 Q6

Video Solution:

Solution:

Solution locked