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mathsVector AlgebraProperties of Cross Product

Understanding Properties of Cross Product

In this page, you will learn about the fundamentals of understanding properties of vector cross 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 xx vec q`
 `quad quad = |(i, j, k),(p_x, p_y, p_z),(q_x, q_y, q_z)|`

Where `p_x, p_y, p_z, q_x, q_y, q_z in RR `
What is the result `vec p xx vec q`?

  • a scalar
  • a vector
  • a vector

The answer is 'a vector'

`vec p xx vec q = |(i, j, k),(p_x, p_y, p_z),(q_x, q_y, q_z)| `
`p_x, p_y, p_z, q_x, q_y, q_z in RR `

What are the x, y, z components of the result?

  • numerical expressions of real numbers
  • numerical expressions of real numbers
  • not numerical expressions as the terms are not numbers

The answer is 'numerical expressions of real numbers'

`vec p xx vec q = |p||q|sin theta hat n`

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

  • a numerical expression of real numbers
  • a numerical expression of real numbers
  • not a numerical expression as the terms are not numbers

The answer is 'a numerical expression of real numbers'

Cross product is a vector with numerical expression as components.

Cross Product as Numerical Expressions: `vec p xx vec q = |(i, j, k),(p_x, p_y, p_z),(q_x, q_y, q_z)| `
where `p_x, p_y, p_z, q_x, q_y, q_z in RR `
`vec p xx vec q = |p||q|sin theta `
where `|p|, |q|, sin theta in RR `
The cross product is a numerical expression of real numbers.

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

 •  Closure Law

 •  Commutative Law

 •  Associative Law

 •  Distributive Law

 •  Modulus in cross product

In learning these, which of the following would help?

  • Memorize each of the laws
  • use the properties of real numbers to understand the cross product as numerical expression
  • use the properties of real numbers to understand the cross product as numerical expression

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

Properties of cross product are understood from properties of real numbers applied to the numerical expression representing the cross product.

Properties of Cross Product: `vec p xx vec q = |(i, j, k),(p_x, p_y, p_z),(q_x, q_y, q_z)| ` is considered as numerical expressions and properties of real numbers are applied to understand properties of cross product.

Solved Exercise Problem:

To understand properties of cross 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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