Vector Arithmetic — Question 10

Question 10

Find constants $a,b$ so that $a\langle1,1,0\rangle+b\langle0,1,1\rangle=\langle2,5,3\rangle$, or prove it is impossible.

Question 10 – Solution

Write every vector in matching component order. Perform scalar multiplication before addition or subtraction, then interpret the resulting components in the context of the problem.

The linear combination is $\langle a,a+b,b\rangle$.

The first and third components force $a=2$ and $b=3$.

Then $a+b=5$, matching the middle component. Hence $\boxed{a=2,\ b=3}$.

The result follows from the defining vector formulas used above, and each component, magnitude, or scalar condition has been checked against the information in the question.

Vector equations behave like ordinary algebra provided that every operation is performed component by component. A final component check is often the fastest verification.