Vectors - The Basics — Question 7

Question 7

A particle moves from $(1,1,1)$ in the direction $\langle2,-2,1\rangle$. At what point has it traveled exactly $12$ units?

Question 7 – 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 direction magnitude is $\sqrt{4+4+1}=3$, so its unit vector is $\frac13\langle2,-2,1\rangle$.

A $12$-unit displacement is $12\widehat v=\langle8,-8,4\rangle$.

Adding the starting point gives $\boxed{(9,-7,5)}$.

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 $1$.