Find a unit vector perpendicular to the plane containing vectors P = 3i − 2j + 4k and Q = i + j − 2k.

Question

ScanSolve AI Tutor · Accepted Answer

To find a unit vector perpendicular to the plane containing two vectors, we first calculate their cross product, and then divide the resulting vector by its magnitude.

Given vectors:
$\vec{P} = 3\vec{i} - 2\vec{j} + 4\vec{k}$
$\vec{Q} = \vec{i} + \vec{j} - 2\vec{k}$

Calculate the cross product $\vec{N} = \vec{P} \times \vec{Q}$ .
$\vec{N} = \begin{vmatrix} \vec{i} & \vec{j} & \vec{k} \\ 3 & -2 & 4 \\ 1 & 1 & -2 \end{vmatrix}$ $\vec{N} = \vec{i}((-2)(-2) - (4)(1)) - \vec{j}((3)(-2) - (4)(1)) + \vec{k}((3)(1) - (-2)(1))$ $\vec{N} = \vec{i}(4 - 4) - \vec{j}(-6 - 4) + \vec{k}(3 - (-2))$ $\vec{N} = 0\vec{i} - \vec{j}(-10) + \vec{k}(5)$ $\vec{N} = 10\vec{j} + 5\vec{k}$
Calculate the magnitude of $\vec{N}$ .
$|\vec{N}| = \sqrt{(0)^2 + (10)^2 + (5)^2}$ $|\vec{N}| = \sqrt{0 + 100 + 25}$ $|\vec{N}| = \sqrt{125}$ $|\vec{N}| = \sqrt{25 \times 5}$ $|\vec{N}| = 5\sqrt{5}$
Find the unit vector $\hat{n}$ .
A unit vector perpendicular to the plane can point in two opposite directions, so we include $\pm$ .
$\hat{n} = \pm \frac{\vec{N}}{|\vec{N}|} = \pm \frac{10\vec{j} + 5\vec{k}}{5\sqrt{5}}$ $\hat{n} = \pm \left( \frac{10}{5\sqrt{5}}\vec{j} + \frac{5}{5\sqrt{5}}\vec{k} \right)$ $\hat{n} = \pm \left( \frac{2}{\sqrt{5}}\vec{j} + \frac{1}{\sqrt{5}}\vec{k} \right)$ $\hat{n} = \boxed{\text{(B) } \pm \frac{1}{\sqrt{5}}(2\vec{j} + \vec{k})}$

Find a unit vector perpendicular to the plane containing vectors P = 3i − 2j + 4k and Q = i + j − 2k.

Answer

Step-by-step solution

Calculate the cross product $\vec{N} = \vec{P} \times \vec{Q}$ .

Calculate the magnitude of $\vec{N}$ .

Find the unit vector $\hat{n}$ .

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More Mathematics Questions

Find a unit vector perpendicular to the plane containing vectors P = 3i − 2j + 4k and Q = i + j − 2k.

Answer

Step-by-step solution

Calculate the cross product N⃗=P⃗×Q⃗\vec{N} = \vec{P} \times \vec{Q}N=P×Q​.

Calculate the magnitude of N⃗\vec{N}N.

Find the unit vector n^\hat{n}n^.

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More Mathematics Questions

Calculate the cross product $\vec{N} = \vec{P} \times \vec{Q}$ .

Calculate the magnitude of $\vec{N}$ .

Find the unit vector $\hat{n}$ .