Vector Arithmetic — Question 6

Question 6

Write $\vec w=\langle7,1\rangle$ as a linear combination of $\vec a=\langle1,2\rangle$ and $\vec b=\langle3,-1\rangle$.

Question 6 – 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.

Let $\vec w=c\vec a+d\vec b$. Components give $c+3d=7$ and $2c-d=1$.

From the second equation $d=2c-1$. Substitute: $c+3(2c-1)=7$, so $7c=10$.

Thus $c=10/7$ and $d=13/7$, giving $\boxed{\vec w=\frac{10}{7}\vec a+\frac{13}{7}\vec b}$.

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.