Q73 / 160 · TS EAMCET 2020 (11 Sep Shift 1) · Mathematics

Let $\mathbf{p}, \mathbf{q}, \mathbf{r}$ be three non-coplanar vectors and $=\mathbf{p} \times \mathbf{q}$. If $\mathbf{a}, \mathbf{b}, \mathbf{c}$ denote the coterminous edges of a parallelopiped, then its height with the base having $\mathbf{a}$ and $\mathbf{c}$ is

  • A
    $|\mathbf{p}|$
  • B
    $\frac{1}{|\mathbf{a}|}$
  • C
    $\frac{1}{|\mathbf{b}|}$
  • Correct answer
    D
    $\frac{1}{|\mathbf{q}|}$

Solution

(d) For three non-coplanar vector $\mathbf{p}, \mathbf{q}$ and $\mathbf{r}$, it is given that $\left[\begin{array}{lll}\mathbf{p} & \mathbf{q} & \mathbf{r}\end{array}\right] \mathbf{a}=\mathbf{q} \times \mathbf{r}$ $\left[\begin{array}{lll}\mathbf{p} & \mathbf{q} & \mathbf{r}\end{array}\right] \mathbf{b}=\mathbf{r} \times \mathbf{p}$ and $\quad\left[\begin{array}{lll}\mathbf{p} & \mathbf{q} & \mathbf{r}\end{array}\right] \mathbf{c}=\mathbf{p} \times \mathbf{q}$ $\therefore \mathbf{a}, \mathbf{b}, \mathbf{c}$ are reciprocal system vectors of vectors $\mathbf{p}, \mathbf{q}, \mathbf{c}$, so $\left[\begin{array}{lll}\mathbf{a} & \mathbf{b} & \mathbf{c}\end{array}\right] \mathbf{p}=\mathbf{b} \times \mathbf{c}$ $\ldots(i)$ $\left[\begin{array}{lll}\mathbf{a} & \mathbf{b} & \mathbf{c}\end{array}\right] \mathbf{q}=\mathbf{c} \times \mathbf{a}$ $\ldots(ii)$ and $\left[\begin{array}{lll}\mathbf{a} & \mathbf{b} & \mathbf{c}\end{array}\right]\mathbf{r}=\mathbf{a} \times \mathbf{b}$ $\ldots(iii)$ $\because \mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ are coterminous edges of $\mathbf{a}$ parallelopiped, so height with the base have a and c is $\frac{[\mathbf{a} \quad \mathbf{b} \mathbf{c}]}{|\mathbf{a} \times \mathbf{c}|}=\frac{1}{|\mathbf{q}|}$ [From Eq. (ii)]

Hence, the correct answer is option: 'D'.

Relevant Questions