# Properties of cross and dot products

• bobey
In summary: Your argument is correct. The given information implies that the vector (v-w) is both perpendicular and parallel to u, which can only happen if (v-w) is the zero vector. Therefore, v and w must be equal.
bobey

## Homework Statement

let u be a nonzero vector in space and let v and w be any two vectors in space. if u.v = u.w and u x v = u x w, can you conclude that v=w? give reason for your answer.

## The Attempt at a Solution

v is not necessary equal to w
since u is nonzero vector thus the cross product of the vector u with v and w will produce another vector and it is consider the same theirs magnitude and directions are same and their dot product will produce a scalar which has no indication of direction. thus v and w is not necessary same vector.

is my argument is true... or can anyone improve it and prove it with example... your help is highly appreciated... thanx

bobey said:

## Homework Statement

let u be a nonzero vector in space and let v and w be any two vectors in space. if u.v = u.w and u x v = u x w, can you conclude that v=w? give reason for your answer.

## The Attempt at a Solution

v is not necessary equal to w
since u is nonzero vector thus the cross product of the vector u with v and w will produce another vector and it is consider the same theirs magnitude and directions are same and their dot product will produce a scalar which has no indication of direction. thus v and w is not necessary same vector.

is my argument is true... or can anyone improve it and prove it with example... your help is highly appreciated... thanx
I have no clue what that means. You start by saying v is not necessarily the same as w, say a couple of general things about cross product and dot product and conclude that v and w are not necessarily the same. What, in what you said about cross product and dot product, led to that conclusion? When you say " it is consider the same their magnitude and directions are same" what vectors are you talking about having the same magnitude and direction? What does "their" refer to?
I think you would be better off using specific formulas. In particular I would recommend using the fact that $u\cdot v= |u||v|cos(\theta)$ and $|u\times v|= |u||v| sin(\theta)$.

It will help to know that for $\theta$ and $\phi$ both between 0 and 180 degrees, $tan(\theta)= tan(\phi)$ implies $\theta= \phi$.

HallsofIvy said:
I have no clue what that means. You start by saying v is not necessarily the same as w, say a couple of general things about cross product and dot product and conclude that v and w are not necessarily the same. What, in what you said about cross product and dot product, led to that conclusion? When you say " it is consider the same their magnitude and directions are same" what vectors are you talking about having the same magnitude and direction? What does "their" refer to?
I think you would be better off using specific formulas. In particular I would recommend using the fact that $u\cdot v= |u||v|cos(\theta)$ and $|u\times v|= |u||v| sin(\theta)$.

It will help to know that for $\theta$ and $\phi$ both between 0 and 180 degrees, $tan(\theta)= tan(\phi)$ implies $\theta= \phi$.

why you think to use this fact : $u\cdot v= |u||v|cos(\theta)$ and $|u\times v|= |u||v| sin(\theta)$ in your arguments? I CAN'T SEE IT!

this is my new argument ::

yes v = w.
u.v = v.u and u x v = u x w implies that u.(v-w)=0 and u x (v-w) = 0 implies that u perpendicular with (v-w) and u parallel to (v-w) implies that u = 0 or (v-w) = 0

thus v=w since u is not a zero vectorIS MY ARGUMENT CORRECT NOW? or any other ideas...THANX 4 D RESPONSE HallsofIvy...

Yes, that is right.

## What are the cross and dot products?

The cross and dot products are mathematical operations used to find the relationship between two vectors in a three-dimensional space.

## What is the difference between the cross and dot products?

The cross product results in a vector that is perpendicular to both of the original vectors, while the dot product results in a scalar value.

## How are the cross and dot products calculated?

The cross product is calculated by taking the magnitude of the two vectors and the sine of the angle between them. The dot product is calculated by taking the magnitude of the two vectors and the cosine of the angle between them.

## What are the applications of cross and dot products?

The cross product is used in physics and engineering to calculate torque and angular momentum, while the dot product is used in physics and mathematics to find the angle between two vectors.

## Can the cross and dot products be used in higher dimensions?

Yes, the cross product can be extended to higher dimensions, while the dot product is already defined for any number of dimensions. However, the interpretation and applications may differ in higher dimensions.

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