Let be the projection vector of , , on the vector . If , then the area of the parallelogram formed by the vectors and is __________. [2025]

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Solution

Q.
Let $\vec{c}$ be the projection vector of $\vec{b} = λ \hat{i} + 4 \hat{k}$ , $λ > 0$ , on the vector $\vec{a} = \hat{i} + 2 \hat{j} + 2 \hat{k}$ . If $| \vec{a} + \vec{c} |$ $= 7$ , then the area of the parallelogram formed by the vectors $\vec{b}$ and $\vec{c}$ is __________. [2025]

Ans.
16

Here, $\vec{c} = (\frac{\vec{b} \cdot \vec{a}}{| \vec{a} |}) \frac{\vec{a}}{| a |} = (\frac{λ + 8}{9}) \vec{a}$

Now, $| \vec{a} + \vec{c} |$ $=$ $| \vec{a} + (\frac{λ + 8}{9}) \vec{a} |$ $= 7$ [Given]

$\Rightarrow$ $| \vec{a} |$ $| \frac{λ + 17}{9} |$ $= 7 \Rightarrow$ $| \frac{λ + 17}{9} |$ $\times 3 = 7 \Rightarrow$ $| \frac{λ + 17}{3} |$ $= 7$

$\Rightarrow$ $| λ + 17 |$ $= 21$

Since, $λ > 0 \Rightarrow λ = 4$

$∴ \vec{c} = \frac{4}{3} \vec{a} = \frac{4}{3} (\hat{i} + 2 \hat{j} + 2 \hat{k}), \vec{b} = 4 (\hat{i} + \hat{k})$

Now, $\vec{b} \times \vec{c} = \frac{16}{3}$ $| \begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 0 & 1 \\ 1 & 2 & 2 \end{matrix} |$ $= \frac{16}{3} (- 2 \hat{i} - \hat{j} + 2 \hat{k})$

$∴$ Area of parallelogram $| \vec{b} \times \vec{c} |$ $= \frac{16}{3} \times 3 = 16$ sq. units.

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