Let a ? = i ^ – 3 j ^ + 7 k ^ , b ? = 2 i ^ – j ^ + k ^ and c ? be a vector such that ( a ? + 2 b ? ) × c ? = 3 ( c ? × a ? ) . If a ? · c ? = 130 , then b ? · c ? is equal to [2024]

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Solution

Q.
Let $\vec{a} = \hat{i} - 3 \hat{j} + 7 \hat{k}$ , $\vec{b} = 2 \hat{i} - \hat{j} + \hat{k}$ and $\vec{c}$ be a vector such that $(\vec{a} + 2 \vec{b}) \times \vec{c} = 3 (\vec{c} \times \vec{a})$ . If $\vec{a} \cdot \vec{c} = 130$ , then $\vec{b} \cdot \vec{c}$ is equal to __________. [2024]

Ans.
(30)

Let $\vec{c} = x \hat{i} + y \hat{j} + z \hat{k}$

Now, $\vec{a} + 2 \vec{b} = 5 \hat{i} - 5 \hat{j} + 9 \hat{k}$

Given, $(\vec{a} + 2 \vec{b}) \times \vec{c} = 3 (\vec{c} \times \vec{a})$

$\Rightarrow$ $| \begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ 5 & - 5 & 9 \\ x & y & z \end{matrix} |$ $= 3$ $| \begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ x & y & z \\ 1 & - 3 & 7 \end{matrix} |$

$\Rightarrow \hat{i} (- 5 z - 9 y) - \hat{j} (5 z - 9 x) + \hat{k} (5 y + 5 x)$

$\Rightarrow 3$ $[\hat{i} (7 y + 3 z) - \hat{j} (7 x - z) + \hat{k} (- 3 x - y)]$

$\Rightarrow - 5 z - 9 y = 21 y + 9 z and 5 z - 9 x = 21 x - 3 z$

$\Rightarrow - 30 y = 14 z and 8 z = 30 x$

$\Rightarrow z = \frac{- 30}{14} y and z = \frac{30}{8} x$

$\Rightarrow z = \frac{- 15}{7} y = \frac{15}{4} x$

$\Rightarrow - 4 y = 7 x$

Now, $\vec{a} \cdot \vec{c} = 130$

$\Rightarrow x - 3 y + 7 z = 130$

$\Rightarrow x - 3 (- \frac{7}{4} x) + 7 (\frac{15}{4} x) = 130$

$\Rightarrow 130 x = 4 \times 130 \Rightarrow x = 4, y = - 7 and z = 15$

$∴ \vec{b} \cdot \vec{c} = (2 \hat{i} - \hat{j} + \hat{k}) (4 \hat{i} - 7 \hat{j} + 15 \hat{k})$

= 8 + 7 +15 = 30.

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