Vectors - The Basics — Question 9

Question 9

Find the point one-third of the way from $A=(7,-2,4)$ to $B=(-2,10,1)$.

Question 9 – Solution

Identify the initial and terminal information first. Build the vector by subtracting coordinates, then use the magnitude formula or normalize only after the components are correct.

The displacement from $A$ to $B$ is $B-A=\langle-9,12,-3\rangle$.

One-third of this displacement is $\langle-3,4,-1\rangle$.

Starting at $A$ gives $A+\frac13(B-A)=\boxed{(4,2,3)}$.

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.

A vector separates two ideas: its components describe displacement, while its magnitude and unit vector describe size and direction. Always check signs and confirm that a unit vector has magnitude $1$.