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

1 Answer
Jan 20, 2016

Answer:

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)#