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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.

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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

click on the content to continue..

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? 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. 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. 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.

slide-show version coming soon