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Thought-Process to Discover Knowledge

Welcome to nubtrek.

Books and other education websites provide "matter-of-fact" knowledge. Instead, nubtrek provides a thought-process to discover knowledge.

In each of the topic, the outline of the thought-process, for that topic, is provided for learners and educators.

Read in the blogs more about the unique learning experience at nubtrek.continue
mathsProperties of Vector ArithmeticsProperties of Cross Product

When products of two vectors are equal

In this page, you will learn the property when two vector cross products are equal



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Consider real numbers `a, b in RR` and an unknown `x`. Given that `ax = ab`, what is `x`?

  • `x=a`
  • `x=b`
  • `x=b`
  • `x=ab`

The answer is '`x=b`'. Both the left hand side and right hand side of the equation is divided by `a` to arrive at the solution.

Consider vectors `vec p, vec q in bbb V` and an unknown `vec x`. Given that `vec x xx vec p = vec q xx vec p`. What is the value of `vec x`?vector cross product no cancellation Note: All the vectors shown in yellow-dotted-line will form parallelograms of same area with `vec p`.

  • cannot be calculated
  • cannot be calculated
  • `vec x = vec q`

The answer is 'Cannot be calculated'.

`vec x xx vec p = vec q xx vec p` does not imply that `vec x = vec q`.
That is, `vec p` cannot be canceled on left-hand-side and right-hand-side. vector cross product no cancellation Note that in cross product, `vec q` is split into orthogonal components and the component in perpendicular to `vec p` is in the product. The component parallel to `vec p` is lost.

 •  Given cross products are equal, does not imply the vectors are equal.

Cannot Cancel: Given `vec x xx vec p = vec q xx vec p` does not imply `vec x = vec q`. That is, the `vec p` cannot be canceled on both sides of the equation or on numerator and denominator in a division.

`vec x xx vec p = vec q xx vec p` imply that the component perpendicular to `vec p` of both `vec x` and `vec q` are equal.vector cross product no cancellation If we subtract `vec x - vec q` then the common component will cancel out and the remaining vector will be parallel to `vec p`.

`vec x xx vec p = vec q xx vec p`

Subtracting `vec q xx vec p` from both the sides. `vec x xx vec p - vec q xx vec p = vec q xx vec p - vec q xx vec p`
`(vec x - vec q)xx vec p = (vec q - vec q)xx vec p`
`(vec x - vec q)xx vec p = 0 xx vec p`
`(vec x - vec q)xx vec p = 0 `

The above can be understood as vector `vec x - vec q` is parallel to `vec p`.

 •  If cross products are equal then difference of the vectors will be collinear to the vector with which cross products are equal.

Subtraction on sides of an Equation: `vec x xx vec p = vec q xx vec p` imply that
`(vec x - vec q)xx vec p = 0`
Which implies `vec x - vec q` is either `vec 0` or is parallel or collinear to `vec p`.

                            
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