Referred to the origin $O$, a variable point $R$ has position vector $\mathbf{r}$ and the fixe

Question

Referred to the origin $O$, a variable point $R$ has position vector $\mathbf{r}$ and the fixed points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$ respectively, where $\mathbf{a}$ and $\mathbf{b}$ are non-zero vectors. If $\mathbf{a}$ and $\mathbf{b}$ are parallel, find $\mathbf{a}\times \mathbf{b}$.[1] If $\mathbf{a}$ and $\mathbf{b}$ are non-parallel, interpret geometrically the following vector equations: $\mathbf{r}\times \left( \mathbf{b}-\mathbf{a} \right)=\mathbf{0}$,[2] $\mathbf{r}\cdot \left( \mathbf{b}-\mathbf{a} \right)=0$.[2] It is now given that the coordinates of $A$ and $B$ are $\left( -1,2,2 \right)$ and $\left( 6,-2,-4 \right)$ respectively. Use a vector product to find the area of the triangle $OAB$.[3]

Accepted Answer

(a) $\mathbf a\times\mathbf b=\mathbf0$. (b)(i) A line through $O$ parallel to $AB$. (b)(ii) A plane through $O$ with normal $\overrightarrow{AB}$. (c) $3\sqrt5\text{ units}^2$.

2024 HCI Promo Q9

2024 HCI Promo Q9

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