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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!
(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!