Vector Arithmetic — Question 7

Question 7

The diagonals of a parallelogram are $\vec d_1=\langle8,2\rangle$ and $\vec d_2=\langle-2,6\rangle$. Find possible adjacent side vectors.

Question 7 – 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.

If adjacent sides are $\vec a,\vec b$, then $\vec d_1=\vec a+\vec b$ and $\vec d_2=\vec a-\vec b$.

Add and divide by two: $\vec a=(\vec d_1+\vec d_2)/2=\langle3,4\rangle$.

Subtract and divide by two: $\vec b=(\vec d_1-\vec d_2)/2=\boxed{\langle5,-2\rangle}$. Thus one ordered pair is $\boxed{\langle3,4\rangle,\langle5,-2\rangle}$.

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.