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summary of this topic

Properties of Cross Product

Properties of Cross Product

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 »  equal vector cross product does not imply vectors are equal

    →  `vec p xx vec q ` ` = vec r xx vec q ` does not imply that `vec p = vec r`

    →  on both sides of the equation, `vec q` cannot be canceled.

    →  `vec p xx vec q ` ` = vec r xx vec q ` imply that `(vec p - vec r) xx vec q = 0`

When products of two vectors are equal

plain and simple summary

nub

plain and simple summary

nub

dummy

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

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

simple steps to build the foundation

trek

simple steps to build the foundation

trek

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


Keep tapping on the content to continue learning.
Starting on learning "When products of two vectors are equal". ;;

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

  • `x=a`
  • `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
  • `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.

`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`.

comprehensive information for quick review

Jogger

comprehensive information for quick review

Jogger

dummy

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.

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`.



           

practice questions to master the knowledge

Exercise

practice questions to master the knowledge

Exercise

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consider real numbers a, b in real numbers, and an unknown real number x. Given that a x = a b, what is x?
1
2
3
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 p, q, in vector space v and an unknown vector x. Given that vector x cross vector p = vector q cross vector p. What is the value of vector x?
cannot;calculated;not
cannot be calculated
vector;x;q
vec x = vec q
The answer is 'Cannot be calculated'.
vector x cross vector p = vector q cross vector p, does not imply that vector x = vector q. That is vector p cannot be canceled on left hand side and right hand side. ;; note that in cross product, vector q is split into orthogonal components and the component in perpendicular to vector p is in the product. The component parallel to vector p is lost.
Given cross products are equal, does not imply the vectors are equal.
Cannot Cancel: Given vector x cross vector p = vector q cross vector p, does not imply vector x = vector q. That is the vector p cannot be canceled on both sided of the equation or on numerator and denominator in a division.
Vector x cross vector p = vector q cross vector p, imply that the component perpendicular to vector p of both vector x and vector q are equal. If we subtract vector x minus vector q, then the common component will cancel out and the remaining vector will be parallel to vector p.
The given proof is understood as vector x minus vector q is parallel to vector 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: Vector x cross vector p = vector q cross vector p imply that vector x minus vector q cross vector p = 0. Which implies vector x minus vector q is either vector 0 or is parallel or collinear to vector p.

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