Let a and b be two vectors. Let | a | = 1 , | b | = 4 and a · b = 2 . If c = ( 2 a × b ) - 3 b , then the value of b · c is [2023]

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Solution

Q.
Let $\vec{a}$ and $\vec{b}$ be two vectors. Let $| \vec{a} |$ $= 1,$ $| \vec{b} |$ $= 4$ and $\vec{a} \cdot \vec{b} = 2 .$ If $\vec{c} = (2 \vec{a} \times \vec{b}) - 3 \vec{b},$ then the value of $\vec{b} \cdot \vec{c}$ is [2023]

1 - 60

2 - 48

3 - 84

4 - 24

Ans.
(2)

$We have$ $| \vec{a} |$ $= 1,$ $| \vec{b} |$ $= 4,$ $\vec{a} \cdot \vec{b} = 2$

$\vec{c} = (2 \vec{a} \times \vec{b}) - 3 \vec{b}$

$\vec{b} \cdot \vec{c} = \vec{b} \cdot ((2 \vec{a} \times \vec{b}) - 3 \vec{b}) = \vec{b} \cdot (2 \vec{a} \times \vec{b}) - 3 \vec{b} \cdot \vec{b}$

$= 0 - 3 | \vec{b} |^{2} = - 3 \times {(4)}^{2} = - 3 \times 16 = - 48$

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