Question 06 (a)
Consider the vectors
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?
If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.
Check whether the vectors are pairwise orthogonal.
The four vectors will form a basis of if they are linearly independent.
Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.
First we must determine whether the vectors are orthogonal. We know that two vectors are orthogonal if their dot product is zero.
Consider . This is equal to
All other dot products are calculated in the same way and all vanish. Hence is an orthogonal set.
Since an orthogonal set of nonzero vectors must be linearly independent, and a set of 4 linearly independent vectors in must be a basis of , we conclude that is an orthogonal basis of .