What is the solution to this vector problem?

Question

Find the vector b co-linear with the vector a = (2, 1, −1) and satisfying the condition b ◦ a = 3.

How to approach this kind of problems? What is the solution?

Answer 1

To be collinear the vectors have to be parallel. But that means there is a nonzero number $c$ so that $\mathbf{b}= c \mathbf{a}= \langle 2c,c,-c \rangle$. But we know that the dot product must be $3$ so that $$ 3=\mathbf{b} \cdot \mathbf{a}= \langle 2c,c,-c \rangle \cdot \langle 2,1,-1 \rangle= 4c+c+c= 6c $$ But then $c=1/2$, implying that $b= \langle 1,1/2,-1/2 \rangle$.

Answer 2

HINT

If $b$ is co-linear with $a$, it is just a scale of the elements of $a$, in other words, $$ \vec{b} = x \vec{a} = x(2,1,-1) = (2x,x,-x), $$ for some $x \in \mathbb{R}$.

Can you now explicitly compute $\vec{a} \cdot \vec{b}$ and solve for $x$?