If $\mathbf{p} \times \mathbf{q}=\mathbf{p} \times \mathbf{r}$ and $\mathbf{p} \cdot \mathbf{q}=\mathbf{p}…

If $\mathbf{p} \times \mathbf{q}=\mathbf{p} \times \mathbf{r}$ and $\mathbf{p} \cdot \mathbf{q}=\mathbf{p} \cdot \mathbf{r}$, then $\ldots . . .$.
  1. $p=r$
  2. $q=r$
  3. $p=q$
  4. $p+q=0$

Solution

$\mathbf{p} \times \mathbf{q}=\mathbf{p} \times \mathbf{r}$ $ \begin{aligned} & \Rightarrow \quad \mathbf{p} \times(\mathbf{q}-\mathbf{r})=0 \\ & \text { p.q }=\text { p. } r \\ & \Rightarrow \quad \text { p. }(\mathbf{q}-\mathbf{r})=0 \\ & \end{aligned} $ From Eqs. (i) and (ii) we can say that $\mathbf{p}$ is neither parallel nor Perpendicular to $(\mathbf{q}-\mathbf{r})$ $ \begin{aligned} & \Rightarrow \quad \mathbf{q}-\mathbf{r}=0 \\ & {[\because \mathbf{p} \neq 0]} \\ & \Rightarrow \quad \mathbf{q}=\mathbf{r} \\ & \end{aligned} $ Hence, option (2) is correct

Asked in: AP EAMCET 2020 (22 Sep Shift 2)

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