Dot Product — Question 6

Question 6

Find the scalar projection of $\vec a=\langle-2,5,4\rangle$ onto $\vec b=\langle1,2,2\rangle$, and explain the sign.

Question 6 – Solution

Compute the dot product carefully, then choose the formula that matches the question. A zero, positive, or negative dot product also gives immediate geometric information.

Compute $\vec a\cdot\vec b=-2+10+8=16$ and $\|\vec b\|=3$.

The scalar projection is $\operatorname{comp}_{\vec b}\vec a=(\vec a\cdot\vec b)/\|\vec b\|=\boxed{16/3}$.

It is positive because $\vec a$ has a net component in the same direction as $\vec b$.

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.

The dot product converts two vectors into a scalar measuring directional agreement: positive means generally aligned, zero means perpendicular, and negative means generally opposed.