If h is the altitude of a parallelopiped determined by the vectors ^ a , ^ b , ^ c and the base is taken to be the parallelogram determined by ^ a and ^ b where ^ a = ^ i + ^ j + ^ k , ^ b = 2 ^ i + 4 ^ j - ^ k and ^ c

Correct answer: C. 8 Volume of the parallelopiped = |[ ^ a ^ b ^ c ]| ⇒ (area of the base parallelogram) × h = |[ ^ a ^ b ^ c ]| ⇒ | ^ a × ^ b |h = |[ ^ a ^ b ^ c ]| ⇒ |- 5 ^ i + 3 ^ j + 2 ^ k |h = | cases 1 & 1 & 1 2 & 4 & - 1 1 & 1 & 3 cases | ⇒ 38 h = 4 ⇒ h = (4)/(38) ⇒ 38h^ 2 = 4 ^ 2 ⇒ 19h^ 2 = 8 Hence, option (c) is correct.

BITSAT 2018

Exam: BITSAT
Year: 2018
Subject: Mathematics
Chapter: Vector Algebra

BITSAT 2018 · Mathematics · Vector Algebra

If h is the altitude of a parallelopiped determined by the vectors...

h

is the altitude of a parallelopiped determined by the vectors

\overset{^}{a}, \overset{^}{b}, \overset{^}{c}

and the base is taken to be the parallelogram determined by

\overset{^}{a}

and

\overset{^}{b}

where

\overset{^}{a} = \overset{^}{i} + \overset{^}{j} + \overset{^}{k}, \overset{^}{b} = 2\overset{^}{i} + 4\overset{^}{j} - \overset{^}{k}

and

\overset{^}{c} = \overset{^}{i} + \overset{^}{j} + 3\overset{^}{k},

then the value of

19h^{2}

Options

A $19$
B $16$
C $8$
DNone of these

Correct answer

C. $8$

Step-by-step solution

Volume of the parallelopiped $= \left|\left[\overset{^}{a}\overset{^}{b}\overset{^}{c}\right]\right|$

$⇒$ (area of the base parallelogram) $\times h = \left|\left[\overset{^}{a}\overset{^}{b}\overset{^}{c}\right]\right|$

$⇒ \left|\overset{^}{a} \times \overset{^}{b}\right|h = \left|\left[\overset{^}{a}\overset{^}{b}\overset{^}{c}\right]\right|$

$⇒ \left|- 5\overset{^}{i} + 3\overset{^}{j} + 2\overset{^}{k}\right|h = \left|$ \begin{cases}1 & 1 & 1 \\ 2 & 4 & - 1 \\ 1 & 1 & 3\end{cases} $\right|$

$⇒ \sqrt{38}h = 4 ⇒ h = \dfrac{4}{38}$

$⇒ 38h^{2} = {4}^{2} ⇒ 19h^{2} = 8$

Hence, option (c) is correct.

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