# Solving x=(x x A)+B: Cross Product Solution?

• thenewbosco
In summary, the conversation is about solving the equation x=(x x A)+B, where all variables represent vectors. The person is looking for suggestions on how to isolate x by using cross products with different vectors and utilizing the BAC-CAB rule to simplify double cross products. They are advised to experiment with dotting and crossing with different vectors until they are able to solve the equation.
thenewbosco
I am supposed to solve this explicitly for x:

x=(x x A)+B
where all variables represent vectors.

I moved the B vector to the other side and I was thinking i can take a cross product of something with each side to isolate x but do not know what works here, i tried crossing both sides with each of the vectors.
any suggestions on this one?
thanks

any help at all on this one?

Try dotting and crossing with different things (like, x, A, B). Use the BAC-CAB rule to simplify double cross products. Just mess around for a while and you should be able to figure it out.

## 1. What is a cross product?

A cross product is a mathematical operation that takes two vectors and produces a third vector that is perpendicular to the first two.

## 2. How do you solve the equation x=(x x A)+B?

To solve this equation, you can use the properties of cross products. Rewrite the equation as x=(A x x)+B and then take the cross product of both sides with A, giving you (x x A)=(A x A x x). Since A x A is equal to zero, the equation becomes (x x A)=0, which means x and A are parallel. Therefore, the solution can be written as x=kA+B, where k is a constant.

## 3. What is the significance of the cross product in vector algebra?

The cross product is significant in vector algebra because it allows us to find a vector that is perpendicular to two given vectors. This is useful in many applications, such as calculating torque in physics or finding the direction of a magnetic field in electromagnetism.

## 4. Can you give an example of how to use the cross product in real life?

Sure, one example is in 3D computer graphics. The cross product can be used to calculate the normal vector of a surface, which is important for shading and lighting in rendering 3D objects. Without the cross product, it would be much more difficult to accurately represent the angles and shapes of objects in a 3D space.

## 5. Are there any limitations to using the cross product in solving equations?

Yes, the cross product is only defined in three-dimensional space. It also requires the vectors to be perpendicular to each other, otherwise the cross product will be equal to zero. Additionally, the cross product is not commutative, meaning A x B is not necessarily equal to B x A. These limitations must be taken into consideration when using the cross product in solving equations.

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