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

Question

Referred to the origin $O$, the points $A$, $B$ and $C$ have position vectors $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$. $C$ lies on $AB$ such that $AC : AB = 1 : 3$. Find $\mathbf{c}$ in terms of $\mathbf{a}$ and $\mathbf{b}$.[2] Show that the area of triangle $OAC$ is given by $k|\mathbf{a} \times \mathbf{b}|$, where $k$ is a constant to be determined.[2] It is now given that $\mathbf{a}$ and $\mathbf{b}$ are unit vectors. Using the result in part (i), find the value of $(2\mathbf{a} - \mathbf{b}) \cdot \mathbf{c}$.[3]

Accepted Answer

(i) $\mathbf c=\dfrac{2\mathbf a+\mathbf b}{3}$ (ii) $k=\dfrac16$, area $=\dfrac16|\mathbf a\times\mathbf b|$ (iii) $1$

2024 MI PRU1 Q4

2024 MI PRU1 Q4

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