# How to approach a cross product question

• I
• gregi_2
In summary, the conversation discusses a physics problem involving cross products and the steps needed to solve it. The problem involves demonstrating the relationship a x (b x c) = (a · c)b - (a · b)c and deriving expressions for (a x b) · (c x d) and (a x b)^2 using vectors in terms of their components. The speaker also recommends studying a specific essay to understand the geometric concepts used in physics.
gregi_2
TL;DR Summary
I am hoping for advice on how to approach a what I assume to be cross product related problem that I have never encountered before
I am beginning this new general physics course and I have encountered a question involved with what I assume to be cross products, a topic that I have very little experience with. I am not looking for a direct answer to the problem but advice on what steps should be taken in order to learn how to answer the problem. The problem is as follows,

Demonstrate the following relationship: a x (b x c) = (a · c)b - (a · b)c
Starting from this relationship derive expressions for the following
(a x b) · (c x d)
(a x b)^2

You should write out the vectors in terms of their components! There are quite a few tricks to make the process simpler, like using summation convention, Kroneker deltas and Levi-Cevita symbols, knowing a few cute identities, but you'll get there nonetheless.

gregi_2 said:
I am not looking for a direct answer to the problem but advice on what steps should be taken in order to learn how to answer the problem.
If you find the time, then https://arxiv.org/pdf/1205.5935.pdf is a recommendable essay to study the geometric concepts which are used in physics.

etotheipi

## 1. What is a cross product?

A cross product is a mathematical operation that takes two vectors as input and produces a third vector that is perpendicular to both input vectors. It is commonly used in physics and engineering to calculate the direction and magnitude of a force or torque.

## 2. How do you calculate a cross product?

To calculate a cross product, you can use the formula:
a x b = (aybz - azby)i + (azbx - axbz)j + (axby - aybx)k
where i, j, and k are unit vectors in the x, y, and z directions, and a and b are the two input vectors. Alternatively, you can use the determinant method to calculate the cross product.

## 3. What are the properties of a cross product?

The properties of a cross product include:
- The cross product of two parallel vectors is zero.
- The cross product of two perpendicular vectors is equal to the product of their magnitudes.
- The cross product is anti-commutative, meaning a x b = -b x a.
- The cross product is distributive, meaning a x (b + c) = a x b + a x c.
- The magnitude of the cross product is equal to the area of the parallelogram formed by the two input vectors.

## 4. When should I use a cross product?

A cross product is commonly used in situations where you need to calculate the direction and magnitude of a force or torque, such as in physics and engineering problems. It is also used in vector calculus and in 3D geometry to find angles and distances between vectors.

## 5. What are some common mistakes when approaching a cross product question?

Some common mistakes when approaching a cross product question include:
- Forgetting to take the order of the vectors into account, resulting in a negative or incorrect answer.
- Incorrectly calculating the magnitude of the cross product.
- Using the wrong formula or method to calculate the cross product.
- Not understanding the properties of a cross product and how they can be used to simplify the calculation.

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