Question

Given two vectors a⃗ =−2.00i^+3.00j^+4.00k^ and b⃗ =3.00i^+1.00j^−3.00k^. Obtain a unit vector perpendicular to these two vectors. Express your answer as a unit vector n^ in the form nx, ny, nz where the x, y, and z components are separated by commas.

Answer 1

we are given two vectors as

$a=-2i+3j+4k$

$b=3i+1j-3k$

now, we can find cross product

$aXb=(-2i+3j+4k)X(3i+1j-3k)$

$=\begin{pmatrix}3\left(-3\right)-4\times \:1&4\times \:3-\left(-2\left(-3\right)\right)&-2\times \:1-3\times \:3\end{pmatrix}$

$aXb=-13i+6j-11k$

$|aXb|=\sqrt{(-13)^2+(6)^2+(-11)^2}$

$|aXb|=\sqrt{326}$

now, we can find normal unit vector

$n=\frac{aXb}{|aXb|}$

now, we can plug values

$n=\frac{(-13i+6j-11k)}{\sqrt{326}}$

$n=\frac{-13}{\sqrt{326}}i+\frac{6}{\sqrt{326}}j-\frac{11}{\sqrt{326}}k$

now, we can find components

$n_x=\frac{-13}{\sqrt{326}},n_y=\frac{6}{\sqrt{326}}j,n_z=-\frac{11}{\sqrt{326}}$

...............Answer