How to Test Whether Two Vectors Are Parallel

Quick answer: Two nonzero vectors are parallel when one is a scalar multiple of the other. In two dimensions, (a,b) and (c,d) satisfy ad − bc = 0 when they are parallel.

Worked example

Are (2,−3) and (−4,6) parallel?

1. Multiplying (2,−3) by −2 gives (−4,6).
2. They are parallel but point in opposite directions.
3. The determinant check gives 2×6 − (−3)(−4) = 12 − 12 = 0.

Check the result

The negative multiplier explains the reversed direction without changing the line of action.

Common mistake to avoid

Component ratios can divide by zero. Use scalar multiples or the determinant test; a zero vector has no defined direction and needs separate treatment.

Try it yourself

1. Are (0,2) and (0,5) parallel? 2. Are (1,2) and (2,3) parallel?

Answers and reasoning

1. Yes: multiply by 5/2, avoiding a 0/0 ratio. 2. No: 1×3 − 2×2 = −1, not zero.

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