Given that $\mathbf{p} \cdot \mathbf{q} = 0$, what can be deduced about the vectors \(\mathbf

Question

Given that $\mathbf{p} \cdot \mathbf{q} = 0$, what can be deduced about the vectors $\mathbf{p}$ and $\mathbf{q}$?[2] With respect to the origin on a $xy$-plane, a fixed point $G$ has position vector $\mathbf{g}$ and a variable point $R$ has position vector $\mathbf{r}$. Describe the locus of $R$ if $\mathbf{r} \cdot (\mathbf{r} - \mathbf{g}) = 0$.[2] With respect to the origin, plane $p$ has an equation given by $\mathbf{r} = \mathbf{i} + \mathbf{j} + \mathbf{k} + \lambda(\mathbf{i} - 2\mathbf{j} + 2\mathbf{k}) + \mu(2\mathbf{i} + 3\mathbf{j} - 2\mathbf{k})$, $\lambda \in \mathbb{R}$, $\mu \in \mathbb{R}$. Points $Q$ and $R$ have position vectors $3\mathbf{i} + 9\mathbf{j} + 5\mathbf{k}$ and $4\mathbf{i} + 11\mathbf{j} + 7\mathbf{k}$ respectively. Find the length of the projection of $\overrightarrow{QR}$ onto $p$.[4]

Accepted Answer

(a) (i) ${\mathbf{p}}$ is perpendicular to ${\mathbf{q}}$ or ${\mathbf{p}}$ is a zero vector or ${\mathbf{q}}$ is a zero vector. (a) (ii) The locus of ${R}$ is a circle with diameter ${OG}$. (b) ${\frac{15}{\sqrt{89}}}$

2024 NJC Promo Q7

2024 NJC Promo Q7