2022HSCShort answer

Q14 (a) (i)

2 marks

Question

The two non-parallel vectors $\vec u$ and $\vec v$ satisfy $\lambda\vec u+\mu\vec v=\vec0$ for some real numbers $\lambda$ and $\mu$ . Show that $\lambda=\mu=0$ .

Try the question before revealing the answer.

Solution

Criteria	Marks
Provides correct solution	2
Attempts to use a proof by contradiction or contrapositive, or equivalent merit	1

Sample answer:
Assume $\lambda\vec u+\mu\vec v=\vec0$ .

If $\lambda$ and $\mu$ are both non-zero, then $\vec u=-\frac\mu\lambda\vec v$ and $\vec u,\vec v$ are parallel.

But $\vec u,\vec v$ are not parallel. Hence, one of $\lambda$ or $\mu$ is zero.

Assume the other one is not zero. Without loss of generality, assume $\lambda=0$ but $\mu\ne0$ ; then $\mu\vec v=\vec0$ .

But $\vec v\ne\vec0$ so $\mu=0$ , a contradiction. Hence $\lambda=\mu=0$ .

Supplied marking guidance · human reviewed

Compare this with your working, not only your final sentence.

Syllabus alignment

ProofVectors

The Nature of ProofUse formal proof language, contradiction, counter-examples and rigorous arguments involving numbers and inequalities.
Further operations with three-dimensional vectorsUse vector magnitude, unit vectors and scalar products to solve problems and prove geometric results.

Spot a problem? Send feedback