The angle between vectors a and b is 60 and |a| = 2, |b| = 1. Show that |2a – b| = sqrt (13) and |2a + b| = sqrt (21). Do we have to use dot product? How do we find the magnitude of vector (2a + b)? I received this question from a student from Nanyang Junior College and would like to share this approach with all of you. I hope the diagram explains it clearly.

Thomas says

The answer can be obtained without actually having to draw out the vectors. Just using expansion will do.

(2a – b)•(2a – b) = |2a – b|²

= 4a•a – 4a•b + b•b

= 4|a|² – 4|a||b|cos60° + |b|²

= 4(2)² – 4|2||1|(½) + 1²

= 16 – 4 + 1

= 13

|2a – b| = √13

This is essentially the same as the cosine rule of vectors.