Question

Use the geometric definition of the cross product and the properties of the cross product to make the following calculation.

((j+ k) x j ) x k= ___________

Answer 1

By definition of the cross product,

i × i = j × j = k × k = 0

i × j = k

j × k = i

k × i = j

Also, the cross product is anticommutative. That is, for any two vectors a and b, we have

a × b = - (b × a)

The cross product distributes over sums:

((j + k) × j) × k = ((j × j) + (k × j)) × k

The rest follows from the definition and property mentioned above:

((j + k) × j) × k = (0 + (k × j)) × k

((j + k) × j) × k = (k × j) × k

((j + k) × j) × k = (-(j × k)) × k

((j + k) × j) × k = (-i) × k

((j + k) × j) × k = -(i × k)

((j + k) × j) × k = k × i

((j + k) × j) × k = j