Vectors — An angle between vectors

Question

Find the angle between and .

This practice question was inspired by Question 11(e) in the 2025 NSW HSC Mathematics Extension 1 examination, © NESA. It is an original TestMum variation that practises the same mathematical idea, not an official NESA question.

Set up the solution: write the vector relationship component by component, then use a dot product, magnitude or scalar multiple as required.
Simplify the previous line carefully, keeping exact values where possible.
Simplify the previous line carefully, keeping exact values where possible.
Finish the calculation, then check that the result meets the question’s conditions.
State the final answer clearly in the form the question requested.Final answer:

Common mistake: changing so many features that the new example no longer tests the same mathematical idea.

Verification: The vector lies on the line , which is from the positive x-axis.

Any mark labels are a TestMum study aid, not an official NESA marking allocation.