Find the shortest distance from the origin $O$ to the plane $p$ with equation $2x+3y+6z=10$.[2

Question

Find the shortest distance from the origin $O$ to the plane $p$ with equation $2x+3y+6z=10$.[2] Given that point $A$ lies on plane $p$ and $OA=10$, find the acute angle between the plane $p$ and $OA$.[2] Two non-zero vectors $\mathbf{a}$ and $\mathbf{b}$ are such that $\left| \mathbf{a}\times \mathbf{b} \right|=\sqrt{3}$, $\left| \mathbf{a}-\mathbf{b} \right|=1$ and the angle between the vectors $\mathbf{a}$ and $\mathbf{b}$ is ${{30}^{\circ }}$, find the possible values of $\left| \mathbf{a} \right|$ and $\left| \mathbf{b} \right|$.[5] Two fixed points $C$ and $D$ have position vectors $\mathbf{c}$ and $\mathbf{d}$ respectively. By considering the result $\mathbf{c}\times \mathbf{c}=\mathbf{0}$, give a geometrical interpretation of the vector equation $\mathbf{r}\times \left( \mathbf{d}-\mathbf{c} \right)=\mathbf{c}\times \mathbf{d}$.[3]

Accepted Answer

(a)(i) $\frac{10}{7}\text{ units}$. (a)(ii) $8.2^\circ$ (1 d.p.). (b) $(|\mathbf a|,|\mathbf b|)=(2,\sqrt3)$ or $(\sqrt3,2)$. (c) The line through $C$ in the direction $CD$.

2023 NJC Promo Q5

2023 NJC Promo Q5

Video Solution:

Solution:

Solution locked