**– $ \space a \space = \space (4, \space 7, \space -4), \space b \space = \space (3, \space -1, \space 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**.

\[ \space a \space . \space b \space = \space (4, \space 7, \space -4) \space . \space (3, \space -1, \space 1) \]

\[ \space 4 \space . \space 3 \space + \space 7 \space . \space (-1) \space + \space (-4) \space . \space 1 \]

\[ \space = \space 12 \space – \space 7 \space – \space 4 \]

\[ \space = \space 1 \]

Now** magnitude** is:

\[ \space |a| \space = \space \sqrt{4^2 \space + \space 7^2 \space + \space (-4)^2} \]

\[ \space = \space \sqrt{16 \space + \space 49 \space + \space 16} \]

\[ \space = \space \sqrt{81} \]

\[ \space = \space 9 \]

Now **scalar projection** is:

\[ \space comp_a b \space = \space \frac{a.b}{|a|} \]

**Substituting** the **values** will **result** in:

\[ \space comp_a b \space = \space \frac{1}{9} \]

Now **vector projection** is:

\[ \space comp_a b \space = \space [comp_a b]\frac{a}{|a|} \]

By **substituting values**, we get:

\[ \space = \space \frac{4}{81}, \space \frac{7}{81}, \space – \frac{4}{81} \]

## Numerical Answer

The **scalar projection** is:

\[ \space comp_a b \space = \space \frac{1}{9} \]

And the **vector projection** is:

\[ \space = \space \frac{4}{81}, \space \frac{7}{81}, \space – \frac{4}{81} \]

## Example

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

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

First, we have to find the **dot product**.

\[ \space a \space . \space b \space = \space (4, \space 7, \space -4) \space . \space (3, \space -1, \space -4) \]

\[ \space 4 \space . \space 3 \space + \space 7 \space . \space (-1) \space + \space (-4) \space . \space -4 \]

\[ \space = \space 12 \space – \space 7 \space + \space 16 \]

\[ \space = \space 21 \]

Now** magnitude** is:

\[ \space |a| \space = \space \sqrt{4^2 \space + \space 7^2 \space + \space (-4)^2} \]

\[ \space = \space \sqrt{16 \space + \space 49 \space + \space 16} \]

\[ \space = \space \sqrt{81} \]

\[ \space = \space 9 \]

Now **scalar projection** is:

\[ \space comp_a b \space = \space \frac{a.b}{|a|} \]

**Substituting** the **values** will **result** in:

\[ \space comp_a b \space = \space \frac{21}{9} \]

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

\[ \space comp_a b \space = \space \frac{21}{9} \]