# Vector Identity: Validity Checked

• Grand
In summary, the conversation discusses a vector identity problem from a textbook that asks to check the validity of a given identity. The discussion centers around the implication that if a, b, and c are general vectors satisfying a x c = b x c, does this imply c . a - c . b = c|a-b|. The conversation concludes that the identity is satisfied, but there should be a plus/minus sign before the right hand side since the vectors may have the same or opposite directions. This is supported by a counter-example and a proof using the dot product and the cosine of the angle between vectors.
Grand

## Homework Statement

This is a problem from a textbook, Riley Hobson and Bence 'Mathematical Methods for Physics and Engineering'. It asks to check the validity of a vector identity. If a, b and c are general vectors satisfying a x c = b x c, does this imply c . a - c . b = c|a-b|

2. The attempt at a solution
According to the book the identity is satisfied, but I think that there should be a plus/minus sign before the RHS, since we don't know whether the vectors a-b and c are in the same or opposite directions.

Last edited:
Grand said:
does this imply c . a - c . b = c|a-b|

Nothing implies that because the left side is a scalar and the right side is a vector.

LCKurtz said:
Nothing implies that because the left side is a scalar and the right side is a vector.

How is the right hand side a vector? There are absolute value signs around the only vectors in the expression.

cepheid said:
How is the right hand side a vector? There are absolute value signs around the only vectors in the expression.

The OP stated:
"If a, b and c are general vectors satisfying a x c = b x c, does this imply c . a - c . b = c|a-b|"

That makes the right side the vector c times the scalar |a-b|. That's how.

LCKurtz, thanks for your interest in the topic, I would just like to mention that c means 'the vector c' while c means 'the magnitude of vector c'. The bold notation for vectors is widely used.
I would be grateful if someone has an opinion on the problem. Thanks.

Grand said:
LCKurtz, thanks for your interest in the topic, I would just like to mention that c means 'the vector c' while c means 'the magnitude of vector c'. The bold notation for vectors is widely used.
I would be grateful if someone has an opinion on the problem. Thanks.

Perhaps, if you are using such a convention, you should then have bolded them when you posted "If a, b and c are general vectors".

LCKurtz said:
Perhaps, if you are using such a convention, you should then have bolded them when you posted "If a, b and c are general vectors".

Yes, you're right, they should be bolded, thanks for that.

LCKurtz said:
The OP stated:
"If a, b and c are general vectors satisfying a x c = b x c, does this imply c . a - c . b = c|a-b|"

That makes the right side the vector c times the scalar |a-b|. That's how.

<-- That is all I have to say about that. It was quite clear what the OP meant from the fact that he started using boldface for vectors. Your interpretation of his expression is silly.

Grand said:
LCKurtz, thanks for your interest in the topic, I would just like to mention that c means 'the vector c' while c means 'the magnitude of vector c'. The bold notation for vectors is widely used.
I would be grateful if someone has an opinion on the problem. Thanks.

LCKurtz said:
Perhaps, if you are using such a convention, you should then have bolded them when you posted "If a, b and c are general vectors".

Okay! Are we done splitting hairs now? Ready to actually start helping this guy with his problem?

Grand, one thing I can think of as a starting point is the fact that the dot product is distributive over vector addition, therefore it must be true that:

c·a - c·b = c·(a - b)​

You also know that the definition of the dot product of two vectors is the product of the magnitudes of those two vectors multiplied by the cosine of the angle between them. Does that help?

That's how I did it, and I would also explain why do I think that there should be a +/- sign in the RHS.

Starting with the given relation:

a x c = b x c
a x c - b x c = 0
(a - b) x c = 0
(a - b) x c = 0 implies |a - b||c|sinD=0 where D is the angle between the vectors. Since the vectors are general and therefore non-zero, it follows that sinD=0 and this leaves us with 2 possible values for D - 0 and 180 degrees This means that these vectors have either the same or opposite directions.

Now we continue with:
c . a - c . b = c . (a - b) = |c||a - b| cosD = c|a - b| cosD

However the two values for D give two different values for cosD : -1 and 1, and here's where the plus/minus sign comes from:
c . a - c . b = +/-c|a - b|

Welcome to PF!

Hi Grand! Welcome to PF!

Yes, you're right …

as a simple counter-example, if a = 0 and b = c, then you need the minus.

You can show there must be a - sign allowed by just taking an example:

a=<1,0,0>
b=<1,1,0>
c=<0,1,0>

a x c = b x c = <0,0,1>

a . c - b . c = -1 = -|c||a-b|

Thanks a lot for the immediate help, you're invaluable. I'm really grateful.

## What is "Vector Identity: Validity Checked"?

"Vector Identity: Validity Checked" is a term used in mathematics and physics to describe a vector equation that has been confirmed to be valid, or true. This means that the equation follows the rules and principles of vector algebra and is mathematically accurate.

## Why is it important to check the validity of vector identities?

It is important to check the validity of vector identities because they are used to describe important physical properties and relationships in fields such as mechanics, electromagnetism, and quantum mechanics. If an identity is not valid, it can lead to incorrect calculations and conclusions.

## How is the validity of vector identities checked?

The validity of vector identities is checked using mathematical techniques such as substitution, manipulation, and simplification. These techniques involve replacing variables with specific values, rearranging terms, and reducing the equation to its simplest form to determine if it holds true for all possible values of the variables.

## What happens if a vector identity is found to be invalid?

If a vector identity is found to be invalid, it means that the equation does not accurately describe the relationship between the vectors. This could be due to a mistake in the derivation or a violation of the rules of vector algebra. In such cases, the identity may need to be revised or discarded.

## Are all vector identities checked for validity?

No, not all vector identities are checked for validity. Some identities are considered fundamental and have been proven to be valid through rigorous mathematical proofs. These identities are accepted and used without the need for further validation. However, new or complex vector identities may need to be checked for validity to ensure their accuracy.

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