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Problem : What is the angle θ between the vectors v = (2, 5, 3) and w = (1, - 2, 4)? (Hint: your answer can be left as an expression for cosθ).

To solve this problem, we exploit the fact that we have two different ways of computing the dot product. On the one hand, using the component method, we know that v·w = 2 - 10 + 12 = 4. On the other hand, we know from the geometric method that v·w = | v|| w| cosθ. From the components we can compute | v|2 = 4 + 25 + 9 = 38, and | w|2 = 1 + 4 + 16 = 21. Putting all of these equations together we find that

cosθ = 4/    

Problem : Find a vector which is perpendicular to both u = (3, 0, 2) and v = (1, 1, 1).

We know from the geometric formula that the dot product between two perpendicular vectors is zero. Hence we are looking for a vector (a, b, c) such that if we dot it into either u or v we get zero. This gives us two equations:


3a + 2c=0  
a + b + c=0  

Any choice of a, b, and c which satisfies these equations works. One possible answer is the vector (2, 1, - 3), but any scalar multiple of this vector will also be perpendicular to u and v.
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