Find the unit vector perpendicular to both

Question

ScanSolve AI Tutor · Accepted Answer

To find the unit vector perpendicular to two vectors, we first calculate their cross product, which gives a vector perpendicular to both. Then, we divide this resultant vector by its magnitude to get the unit vector.

From the image, the two vectors appear to be:
$\vec{a} = \vec{i} + 2\vec{j} + \vec{k}$
$\vec{b} = 5\vec{i} - \vec{j} + 2\vec{k}$

Calculate the cross product $\vec{a} \times \vec{b}$ .
$\vec{a} \times \vec{b} = \begin{vmatrix} \vec{i} & \vec{j} & \vec{k} \\ 1 & 2 & 1 \\ 5 & -1 & 2 \end{vmatrix}$ $= \vec{i}((2)(2) - (1)(-1)) - \vec{j}((1)(2) - (1)(5)) + \vec{k}((1)(-1) - (2)(5))$ $= \vec{i}(4 - (-1)) - \vec{j}(2 - 5) + \vec{k}(-1 - 10)$ $= \vec{i}(5) - \vec{j}(-3) + \vec{k}(-11)$ $= 5\vec{i} + 3\vec{j} - 11\vec{k}$
Calculate the magnitude of the cross product $|\vec{a} \times \vec{b}|$ .
Let $\vec{c} = 5\vec{i} + 3\vec{j} - 11\vec{k}$ .
$|\vec{c}| = \sqrt{(5)^2 + (3)^2 + (-11)^2}$ $= \sqrt{25 + 9 + 121}$ $= \sqrt{155}$
Compute the unit vector perpendicular to both $\vec{a}$ and $\vec{b}$ .
The unit vector $\hat{n}$ is given by $\frac{\vec{a} \times \vec{b}}{|\vec{a} \times \vec{b}|}$ .
$\hat{n} = \frac{5\vec{i} + 3\vec{j} - 11\vec{k}}{\sqrt{155}}$ $\hat{n} = \frac{5}{\sqrt{155}}\vec{i} + \frac{3}{\sqrt{155}}\vec{j} - \frac{11}{\sqrt{155}}\vec{k}$
The unit vector perpendicular to both vectors is $\boxed{\frac{5}{\sqrt{155}}\vec{i} + \frac{3}{\sqrt{155}}\vec{j} - \frac{11}{\sqrt{155}}\vec{k}}$ .

Find the unit vector perpendicular to both

Step-by-step solution

Calculate the cross product $\vec{a} \times \vec{b}$ .

Calculate the magnitude of the cross product $|\vec{a} \times \vec{b}|$ .

Compute the unit vector perpendicular to both $\vec{a}$ and $\vec{b}$ .

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More Mathematics Questions

Find the unit vector perpendicular to both

Step-by-step solution

Calculate the cross product a⃗×b⃗\vec{a} \times \vec{b}a×b.

Calculate the magnitude of the cross product ∣a⃗×b⃗∣|\vec{a} \times \vec{b}|∣a×b∣.

Compute the unit vector perpendicular to both a⃗\vec{a}a and b⃗\vec{b}b.

Still stuck on this one?

More Mathematics Questions

Calculate the cross product $\vec{a} \times \vec{b}$ .

Calculate the magnitude of the cross product $|\vec{a} \times \vec{b}|$ .

Compute the unit vector perpendicular to both $\vec{a}$ and $\vec{b}$ .