Find the cross product (7,9,6) x (-4,1,5). Is the resulting vector perpendicular to the given vectors

Expand each vector as linear combinations of the standard basis vectors:

(7, 9, 6) = 7 (1, 0, 0) + 9 (0, 1, 0) + 6 (0, 0, 1)

(-4, 1, 5) = - 4 (1, 0, 0) + (0, 1, 0) + 5 (0, 0, 1)

For brevity, write

i = (1, 0, 0)

j = (0, 1, 0)

k = (0, 0, 1)

Then by definition of the cross product,

i x i = j x j = k x k = (0, 0, 0)

i x j = k

j x k = i

k x i = j

and for any two vectors a and b, we have a x b = - b x a.

Now compute the product:

(7i + 9j + 6k) x (-4i + j + 5k)

= - 28 (i x i) - 36 (j x i) - 24 (k x i)

... + 7 (i x j) + 9 (j x j) + 6 (k x j)

... + 35 (i x k) + 45 (j x k) + 30 (k x k)

= - 36 (-k) - 24 j + 7 k + 6 (-i) + 35 (-j) + 45 i

= 39 i - 59 j + 43 k

which is the same as the vector

(39, - 59, 43)

And yes, this vector is perpendicular to both given vectors.