Vectors - The Basics — Question 4

Question 4

The point $P=(2,-1,4)$ is translated five units in the direction of $\vec a=\langle-2,1,2\rangle$. Find the image point $Q$.

Question 4 – 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.

First normalize: $\|\vec a\|=3$, so $\widehat a=\frac13\langle-2,1,2\rangle$.

A five-unit translation has vector $5\widehat a=\langle-10/3,5/3,10/3\rangle$.

Add it to $P$: $Q=P+5\widehat a=\boxed{\left(-\frac43,\frac23,\frac{22}3\right)}$.

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 $1$.