Question

Find the scalar and vector projections of b onto a.a =(−3, 4)b =(7, 9)

Answer 1

Take into account that the scalar projection is given by:

$comp_a\vec{b}=\frac{\vec{a}\cdot\vec{b}}{a}$

the magnitude of a vector is:

$\begin{gathered} a=\sqrt[]{(-3^{})^2+4^2} \\ a=\sqrt[]{25}=5 \end{gathered}$

The dot product between a and b is:

$\vec{a}\cdot\vec{b}=(-3,4)\cdot(7,9)=-3\cdot7+4\cdot9=-21+36=15$

Then, the scalar projection is:

$\text{comp}_a\vec{b}=\frac{15}{5}=3$

Now, consider that the vector projections is given by:

$\begin{gathered} \text{proj}_a\vec{b}=(\frac{\vec{a}\cdot\vec{b}}{a})\frac{a}{\lvert a\rvert} \\ \text{proj}_a\vec{b}=\frac{3}{5}\cdot(-3,4)=(-\frac{9}{5},\frac{12}{5}) \end{gathered}$

Hence, the answer is:

scalar projection of b onto a = 3

vector projection of b onto a = (-9/5 . 12/5)