Consider the two vectors a⃗ = 2î - 3ĵ + 4k̂ and b⃗ = 5î + qĵ - k̂. Find the unit vector perpendicular to both a⃗ and b⃗ if q = 4.

For q = 4, b⃗ = 5î + 4ĵ - k̂. a⃗×b⃗ = |î ĵ k̂; 2 -3 4; 5 4 -1| = -13î + 22ĵ + 23k̂, and |a⃗×b⃗| = √(169 + 484 + 529) = √(1182). Unit vector perpendicular to both: ±(-13î + 22ĵ + 23k̂)/(√(1182)). ±(-13î + 22ĵ + 23k̂)/(√(1182))

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Consider the two vectors a⃗ = 2î - 3ĵ + 4k̂ and b⃗ = 5î + qĵ - k̂. Find the unit vector…

Consider the two vectors $\vec{a} = 2\hat{i} - 3\hat{j} + 4\hat{k}$ and $\vec{b} = 5\hat{i} + q\hat{j} - \hat{k}$. Find the unit vector perpendicular to both $\vec{a}$ and $\vec{b}$ if $q = 4$.

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Answer

For $q = 4$, $\vec{b} = 5\hat{i} + 4\hat{j} - \hat{k}$. $\vec{a}\times\vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & -3 & 4 \\ 5 & 4 & -1 \end{vmatrix} = -13\hat{i} + 22\hat{j} + 23\hat{k}$, and $|\vec{a}\times\vec{b}| = \sqrt{169 + 484 + 529} = \sqrt{1182}$. Unit vector perpendicular to both: $\pm\dfrac{-13\hat{i} + 22\hat{j} + 23\hat{k}}{\sqrt{1182}}$.

Final answer: $\pm\dfrac{-13\hat{i} + 22\hat{j} + 23\hat{k}}{\sqrt{1182}}$