Find the vector, not with determinants, but by using properties of cross products. k × (i − 4j)?

we know that

i x j = k,

j x k = i,

k x i = j.

Also we know anticommutativity of the cross product:

a x b = – b x a

Then k x i – 4 k x j = j + 4i

Look at the image in link first.

k × (i − 4j)

= k x i – k x 4j

= k x i – 4(k x j)

= j – 4(-i)

= j + 4i

(which we normally wrte as 4i + j)