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Question: What is the unit vector perpendicular to the following vectors \(2\widehat{i} + 2\widehat{j} - \wide...

What is the unit vector perpendicular to the following vectors 2i^+2j^−k^2\widehat{i} + 2\widehat{j} - \widehat{k} and 6i^−3j^+2k^6\widehat{i} - 3\widehat{j} + 2\widehat{k}

A

i^+10j^−18k^517\frac{\widehat{i} + 10\widehat{j} - 18\widehat{k}}{5\sqrt{17}}

B

i^−10j^+18k^517\frac{\widehat{i} - 10\widehat{j} + 18\widehat{k}}{5\sqrt{17}}

C

i^−10j^−18k^517\frac{\widehat{i} - 10\widehat{j} - 18\widehat{k}}{5\sqrt{17}}

D

i^+10j^+18k^517\frac{\widehat{i} + 10\widehat{j} + 18\widehat{k}}{5\sqrt{17}}

Answer

i^−10j^−18k^517\frac{\widehat{i} - 10\widehat{j} - 18\widehat{k}}{5\sqrt{17}}

Explanation

Solution

A→=2i^+2j^−k^\overrightarrow{A} = 2\widehat{i} + 2\widehat{j} - \widehat{k}and B→=6i^−3j^+2k^\overrightarrow{B} = 6\widehat{i} - 3\widehat{j} + 2\widehat{k}

C→=A→×B→=(2i^+2j^−k^)×(6i^−3j^+2k^)\overrightarrow{C} = \overrightarrow{A} \times \overrightarrow{B} = \left( 2\widehat{i} + 2\widehat{j} - \widehat{k} \right) \times \left( 6\widehat{i} - 3\widehat{j} + 2\widehat{k} \right)

\widehat{i} & \widehat{j} & \widehat{k} \\ 2 & 2 & - 1 \\ 6 & - 3 & 2 \end{matrix} \right| = \widehat{i} - 10\widehat{j} - 18\widehat{k}$$ Unit vector perpendicular to both $\overrightarrow{A}$ and $\overrightarrow{B}$ $$= \frac{\widehat{i} - 10\widehat{j} - 18\widehat{k}}{\sqrt{1^{2} + 10^{2} + 18^{2}}} = \frac{\widehat{i} - 10\widehat{j} - 18\widehat{k}}{5\sqrt{17}}$$