Proving distributivity of Dot/Cross product

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IsaacA
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Using the definitions in equations 1.1 and 1.4, and appropriate diagrams, show that the dot product and cross product are distributive;
(a) when the three vectors are coplanar;
(b) in the general case.

Eq. 1.1) A dot B = ABcosθ

Eq. 1.4) A cross B = ABsinθN

This is exactly how my book puts the formulas.

I know how the definition of the dot product is derived, and that it's distributive over vector addition, but I don't understand why they're asking why the three vectors are coplanar. I don't see where the third vector comes into play. I haven't even tried solving this on the cross product side because I know if I don't conceptually grasp the dot product part of it the cross product will only frustrate me.

Here's my attempt at this proof:

Part A: Stared at it for a while trying to figure it out and eventually gave up.

Part B: Broke out the comfort food. Cried a little.

1.2 Is the cross product associative?

(Vector A cross Vector B) cross vector C (equals?) Vector A cross (Vector B cross Vector C)

I know the cross product isn't associate because the order of the cross product determines the direction of the resultant vector, but I feel like there's more to it.

Thank you all so much for your help!
 
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IsaacA said:
Using the definitions in equations 1.1 and 1.4, and appropriate diagrams, show that the dot product and cross product are distributive;
(a) when the three vectors are coplanar;
(b) in the general case.

Eq. 1.1) A dot B = ABcosθ

Eq. 1.4) A cross B = ABsinθN

This is exactly how my book puts the formulas.

I know how the definition of the dot product is derived, and that it's distributive over vector addition, but I don't understand why they're asking why the three vectors are coplanar. I don't see where the third vector comes into play. I haven't even tried solving this on the cross product side because I know if I don't conceptually grasp the dot product part of it the cross product will only frustrate me.

Here's my attempt at this proof:

Part A: Stared at it for a while trying to figure it out and eventually gave up.

Part B: Broke out the comfort food. Cried a little.

1.2 Is the cross product associative?

(Vector A cross Vector B) cross vector C (equals?) Vector A cross (Vector B cross Vector C)

I know the cross product isn't associate because the order of the cross product determines the direction of the resultant vector, but I feel like there's more to it.

Thank you all so much for your help!

The question is asking you to prove [itex] \vec{A}\cdot(\vec{B} + \vec{C}) = \vec{A}\cdot \vec{B} + \vec{A} \cdot \vec{C}, [/itex] and the same type of result using [itex] \times [/itex] instead of [itex] \cdot[/itex] . So, of course there have to be three vectors involved.

RGV
 

1. What is the concept of distributivity in the context of dot/cross product?

Distributivity refers to the property of the dot and cross product operations that states that they follow the rules of distribution over addition and subtraction. In other words, the dot and cross products can be distributed over the terms of a sum or difference of vectors.

2. How is distributivity proven for the dot product?

The distributivity of the dot product can be proven using the following steps:1. Use the definition of the dot product to expand the expression.2. Apply the distributive property of multiplication over addition to the expanded expression.3. Use the commutative property of addition to rearrange the terms.4. Simplify the expression to show that it is equal to the original expression.

3. What is the proof for the distributivity of the cross product?

The distributivity of the cross product can be proven using the following steps:1. Expand both sides of the expression using the definition of the cross product.2. Apply the distributive property of multiplication over addition to the expanded expressions.3. Use the fact that the cross product of any vector with itself is zero to simplify the expression.4. Use the fact that the cross product of any two parallel vectors is also zero to further simplify the expression.5. Show that the two simplified expressions are equal, thus proving the distributivity of the cross product.

4. How does distributivity relate to the properties of dot/cross product?

Distributivity is one of the fundamental properties of the dot and cross product operations. It allows us to manipulate and simplify expressions involving these operations, making them more useful in mathematical and scientific applications. Additionally, distributivity allows for the derivation of other important properties of dot and cross product, such as the associative and commutative properties.

5. Can the distributivity of dot/cross product be extended to higher dimensions?

Yes, the distributivity of dot and cross product can be extended to higher dimensions such as 4D or even n-dimensional spaces. The same proof techniques used for 3D vectors can be applied in higher dimensions to show that distributivity still holds true. However, the specific algebraic expressions and operations may vary in these higher dimensions.

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