What is the unit vector that is orthogonal to the plane containing # (-i + j + k) # and # (3i + 2j - 3k) #?
There are two unit vectors here, depending on your order of operations. They are
When you take the cross product of two vectors, you are calculating the vector that is orthogonal to the first two. However, the solution of
As a quick refresher, a cross-product of
and you get each term by taking the product of the diagonal terms going from left to right, starting from a given unit vector letter (i, j, or k) and subtracting the product of diagonal terms going from right to left, starting from the same unit vector letter:
For the two solutions, lets set:
Let's look at both solutions:
As stated above:
As a flip to the first formulation, take the diagonals again, but the matrix is formed differently:
Notice that the subtractions are flipped around. This is what causes the 'Equal and opposite' form.