Find the scalar and vector projections of b onto a.

– $a=(4,7,-4),b=(3,-1,1)$

The main objective of this question is to find the scalar and vector of one vector onto the other vector.

This question uses the concept of vector and scalar projection. A vector projection is indeed the vector that is made when one vector is broken up into two parts, one of which is parallel to the 2nd vector and the other of which is not while scalar projection is sometimes meant by the term scalar component.

Expert Answer

In this question, we have to find the projection of one vector on the other vector. So first, we have to find the dot product.

$$a.b=(4,7,-4).(3,-1,1)$$

$$4.3+7.(-1)+(-4).1$$

$$=12–7–4$$

$$=1$$

Now magnitude is:

$$|a|=\sqrt{4^2+7^2+(-4{)}^2}$$

$$=\sqrt{16+49+16}$$

$$=\sqrt{81}$$

$$=9$$

Now scalar projection is:

$$com{p}_{a}b=\frac{a.b}{|a|}$$

Substituting the values will result in:

$$com{p}_{a}b=\frac{1}{9}$$

Now vector projection is:

$$com{p}_{a}b=[com{p}_{a}b]\frac{a}{|a|}$$

By substituting values, we get:

$$=\frac{4}{81},\frac{7}{81},–\frac{4}{81}$$

Numerical Answer

The scalar projection is:

$$com{p}_{a}b=\frac{1}{9}$$

And the vector projection is:

$$=\frac{4}{81},\frac{7}{81},–\frac{4}{81}$$

Example

Find the scalar projection of vector $b$ on $a$.

•  $a=(4,7,-4),b=(3,-1,-4)$

First, we have to find the dot product.

$$a.b=(4,7,-4).(3,-1,-4)$$

$$4.3+7.(-1)+(-4).-4$$

$$=12–7+16$$

$$=21$$

Now magnitude is:

$$|a|=\sqrt{4^2+7^2+(-4{)}^2}$$

$$=\sqrt{16+49+16}$$

$$=\sqrt{81}$$

$$=9$$

Now scalar projection is:

$$com{p}_{a}b=\frac{a.b}{|a|}$$

Substituting the values will result in:

$$com{p}_{a}b=\frac{21}{9}$$

Thus the scalar projection of vector $b$ on $a$ is:

$$com{p}_{a}b=\frac{21}{9}$$

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