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Thought-Process to Discover Knowledge

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mathsVector AlgebraProperties of Dot Product

### Understanding Properties of 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