__maths__>__Properties of Vector Arithmetics__>__Properties of Dot Product__### Understanding Properties of Dot Product

In this page, you will learn about the fundamentals of understanding properties of vector dot product.

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Given the following

`vec p = p_x i+p_yj+p_zk `

`vec q = q_x i+q_yj+q_zk `

`vec p cdot vec q = p_xq_x+p_yq_y+p_zq_z `

Where `p_x, p_y, p_z, q_x, q_y, q_z in RR `

Can you guess what will be the result `vec p cdot vec q`?

- a real number
- a real number
- an integer
- a fraction
- a whole number

The answer is 'a real number'

`vec p cdot vec q = p_xq_x+p_yq_y+p_zq_z `

`p_x, p_y, p_z, q_x, q_y, q_z in RR `

What is `p_xq_x+p_yq_y+p_zq_z`?

- a numerical expression with terms as real numbers
- a numerical expression with terms as real numbers
- not a numerical expression as the terms are not numbers

The answer is 'a numerical expression with terms as real numbers'

`vec p cdot vec q = |p||q|cos theta `

What is `|p||q|cos theta`?

- a numerical expression with terms as real numbers
- a numerical expression with terms as real numbers
- not a numerical expression as the terms are not numbers

The answer is 'a numerical expression with terms as real numbers'

Dot product is a numerical expression with terms as real numbers.

**Dot Product as Numerical Expression: ** `vec p cdot vec q = p_xq_x+p_yq_y+p_zq_z `

where `p_x, p_y, p_z, q_x, q_y, q_z in RR `

`vec p cdot vec q = |p||q|cos theta `

where `|p|, |q|, cos theta in RR `

The dot product is a numerical expression of real numbers.

To understand properties of dot product, the following are to be learned

• Closure Law

• Commutative Law

• Associative Law

• Distributive Law

• Modulus in dot product

In learning these, which of the following would help?

- Memorize each of the laws
- use the properties of real numbers to understand the dot product as numerical expression
- use the properties of real numbers to understand the dot product as numerical expression

The answer is 'use the properties of real numbers to understand the dot product as numerical expression'.

Properties of dot product is understood from properties of real numbers applied to the numerical expression representing the dot product.

**Properties of Dot Product: ** `vec p cdot vec q = p_xq_x+p_yq_y+p_zq_z ` is considered as an numerical expression and properties of real numbers are applied to understand properties of dot product.

*Solved Exercise Problem: *

To understand properties of dot product, which of the following number system is used?

- Integers
- Rational numbers
- Real Numbers
- Real Numbers
- None of the above

The answer is 'Real Numbers'

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