What is the unit vector that is orthogonal to the plane containing # ( - 5 i + 4 j - 5 k) # and # (4 i + 4 j + 2 k) #?
There are two steps: (1) find the cross product of the vectors, (2) normalise the resultant vector. In this case, the answer is:
The cross product of two vectors yields a vector that is orthogonal (at right angles) to both.
The cross product of two vectors
First step is to find the cross product:
This vector is orthogonal to both the original vectors, but it is not a unit vector. To make it a unit vector we need to normalise it: divide each of its components by the length of the vector.
The unit vector orthogonal to the original vectors is:
This is one unit vector that is orthogonal to both the original vectors, but there is another - the one in the exact opposite direction. Simply changing the sign of each of the components yields a second vector orthogonal to the original vectors.
(but it's the first vector that you should offer as the answer on a test or assignment!)