Q66 / 180 · KCET 2023 · Mathematics

$|\mathbf{a} \times \mathbf{b}|^2+|\mathbf{a} \cdot \mathbf{b}|^2=144$ and $|\mathbf{a}|=4$, then $|\mathbf{b}|$ is equal to

  • Correct answer
    A
    $3$
  • B
    $8$
  • C
    $4$
  • D
    $12$

Solution

Given, $|\mathbf{a} \times \mathbf{b}|^2+|\mathbf{a} \cdot \mathbf{b}|^2=144$ and $|\mathbf{a}|=4$ $\begin{aligned} & \Rightarrow|\mathbf{a}|^2|\mathbf{b}|^2 \sin ^2 \theta+|\mathbf{a} \| \mathbf{b}|^2 \cos ^2 \theta=144 \\ & \Rightarrow \quad|\mathbf{a}|^2|\mathbf{b}|^2\left[\sin ^2 \theta+\cos ^2 \theta\right]=144\end{aligned}$ $\begin{array}{ll}\Rightarrow \quad & (4)^2 \times|b|^2=144 \\ \Rightarrow \quad & |b|^2=\frac{144}{16}=9 \\ & |b|=3\end{array}$

Hence, the correct answer is option: 'A'.

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