Question

Find the scalar and vector projections of b onto a. a=(4,7,-4),b=(3,-1,1)

Vectors
Find the scalar and vector projections of b onto a.
$$a=(4,7,-4),b=(3,-1,1)$$

2021-05-18
Find the dot product
$$a\cdot b=(4,7,-4)\cdot(3,-1,1)$$
$$=4\cdot3+7\cdot(-1)+(-4)\cdot1$$
$$=12-7-4$$
$$=1$$
Find the magnitude of a
$$|a|=\sqrt{4^2+7^2+(-4)^2}$$
$$=\sqrt{16+49+16}$$
$$=\sqrt{81}$$
$$=9$$
Scalar projection of b on to a is given by
$$\text{comp}_ab=\frac{a\cdot b}{|a|}$$
Substitute the values of the $$a\cdot b$$ and |a|, to get
$$\text{comp}_ab=\frac{1}{9}$$
Vector projection of b onto a is given by
$$\text{proj}_ab=[\text{comp}_ab]\frac{a}{|a|}$$
Substitute the values of the $$\text{comp}_ab$$ and |a|, to get
$$\text{proj}_ab=[\frac{1}{9}]\frac{(4,7,-4)}{9}=(\frac{4}{81},\frac{7}{81},-\frac{4}{81})$$
Result: $$\text{comp}_ab=\frac{1}{9}\quad\text{proj}_ab=(\frac{4}{81},\frac{7}{81},-\frac{4}{81})$$