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Find the cross product and show it is orthogonal to both u and v.

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Find the cross product and show it is orthogonal to both u and v.
u = 6k = <0,0,6>
v = -i + 3j + k = <-1,3,1>

u x v = 6k x (-i + 3j + k)

= -6(k x i) + 18(k x j) + 6(k x k)

= -6j + 18(-i) + 6(0)

= –18i – 6j = <-18,-6,0>

Orthogonal/perpendicular if dot product is 0:

(–18i – 6j) • (6k) = (-18)(0) + (-6)(0) + (0)(6) = 0

(–18i–6j) • (-i+3j+k) = (-18)(-1) + (-6)(3) + (0)(1) = 0

So the cross product u x v is orthogonal to u and v.