Question

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

1. ((i + j) x i) x j =
2. (j + k) x (j x k) =
3. 5i x (i + j) =
4. (k + j) x (k - j) =

Answer 1

The key here is the property of the cross product that identifies it as orthogonal to the 2 vectors. For instance using just the definition of cross product and properties you could say that

i x j = k, since k is orthogonal to both i and j. Also note that i x j = -k is also a possible solution since it is also orthogonal to both i and j. (same line but the vector's direction is opposite).

So for (j+k) x j, you use the property that (j+k) x j = (j x j) + (j x k) and since j x j = 0, you get the vector < 0, 0, 0> + < 1, 0, 0> = < 1, 0, 0> which = i. then you would complete this by taking i x k which of course = j.

You can do similar with the remaining problems and verify your results using the discriminant method.

I hope that answered your question :)