Question

What is the unit vector that is orthogonal to the plane containing `(8i + 12j + 14k)` and `(3i – 4j + 4k)`?

Answer 1

Use the cross product to find the vector `v_1 xx v_2_|_ = v_3`
`v_3 =104i +10j -68k`
`vec(u_3) = 1/sqrt(104^2+10^2+68^2)(104i +10j -68k)`

Explanation:

let `vec(v_1) = (8, 12, 14) and vec(v_2) = (3, -4, 4)`
cross product #vec(v_1)xx vec(v_2) = [(12xx4) +(14xx4)] i + [(14xx3) -(8xx4)] j + [(8xx-4 - 12xx3)]k#

Simplify to get the Orthogonal vector.

`vec(v_3) =104i +10j -68k`
To calculate the unit vector find:
`vec(u_3) = 1/|vec(v_3)| vec(v_3)`
where: `|vec(v_3)| = magnitude`
`|vec(v_3)| = sqrt(104^2+10^2+68^2)`