First slide

Vectors

Question

If $\vec{A} = - 2 \hat{i} + 3 \hat{j} - 4 \hat{k}$ and $\vec{B} = 3 \hat{i} - 4 \hat{j} + 5 \hat{k}$ , then $sinθ =$ is

Moderate

A

$\frac{\sqrt{6}}{\sqrt{29} \times \sqrt{50}}$
B

$\frac{6}{29 \times 50}$
C

$\frac{\sqrt{29} \times \sqrt{50}}{\sqrt{6}}$
D

$\frac{\sqrt{6} \times \sqrt{29}}{\sqrt{50}}$

Solution

$\begin{array}{l} \sin θ = \frac{| \vec{A} \times \vec{B} |}{| \vec{A} | | \vec{B} |} \\ \vec{A} \times \vec{B} = - \hat{i} - 2 \hat{j} - \hat{k} \\ i . e ., | \vec{A} \times \vec{B} | = \sqrt{[(1)^{2} + (2)^{2} + (1)^{2}]} = \sqrt{6} \end{array}$
$\begin{array}{l} | \vec{A} | = \sqrt{[(- 2)^{2} + (3)^{2} + (- 4)^{2}]} = \sqrt{29} \\ | \vec{B} | = \sqrt{[(3)^{2} + (- 4)^{2} + (5)^{2}]} = \sqrt{50} \\ ∴ \sin θ = \frac{\sqrt{6}}{\sqrt{29} \times \sqrt{50}} \end{array}$

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