QJEEM21Maa3f8: JEE Main 2021 Physics - Class 11 JEEM21Maa3f8 Solution: Let and . If the area of the parallelogram whose adjacent sides …

Question

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Let

\overrightarrow a = \widehat i + \alpha \widehat j + 3\widehat k

and

\overrightarrow b = 3\widehat i - \alpha \widehat j + \widehat k

. If the area of the parallelogram whose adjacent sides are represented by the vectors

\overrightarrow a

and

\overrightarrow b

is

8\sqrt 3

square units, then

\overrightarrow a

.

\overrightarrow b

is equal to __________.

Accepted Answer

\overrightarrow a = \widehat i + \alpha \widehat j + 3\widehat k

\overrightarrow b = 3\widehat i - \alpha \widehat j + \widehat k

Area of parallelogram =

\left| {\overrightarrow a \times \overrightarrow b } \right|

= \left| {(\widehat i + \alpha \widehat j + 3\widehat k) \times (3\widehat i - \alpha \widehat j + \widehat k)} \right|

8\sqrt 3 = \left| {(4\alpha )\widehat i + 8\widehat j - (4\alpha )\widehat k} \right|

(64)(3) = 16{\alpha ^2} + 64 + 16{\alpha ^2}

(64)(3) = 32{\alpha ^2} + 64

6 = {\alpha ^2} + 2

{\alpha ^2} = 4

\therefore

\overrightarrow a = \widehat i + \alpha \widehat j + 3\widehat k

\overrightarrow b = 3\widehat i - \alpha \widehat j + \widehat k

\overrightarrow a \,.\,\overrightarrow b = 3 - {\alpha ^2} + 3

= 6 - {\alpha ^2}

= 6 - 4

= 2

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