Distributive Law and Vectors

[unknown101]

Homework Statement

Let Vector a = (3, 4, 1), Vector b = (5, -2, 3) and Vector c = (0, 1, -3). Does Vector a × (Vector b + Vector c) = (Vector a × Vector b) + (Vector a × Vector c)

Homework Equations

a X (b+c)=(a X b)+(a X c)

The Attempt at a Solution

3 5
4 -1
1 0
3 5
4 -1
1 0 Cross out the top 2 numbers and the bottom 2 numbers

Draw 3 arrows going down and three arrows going up. Then subtract down minus up.

for a X (b +c)

(b+c) x=5+0=5
y=-2+1=-1
z=3-3=0
Therefore (b+c=5,-1,0)

a X (b+c)= (3,4,1) X (5,-1,0)
=(15, -4, 0)
According the answer this is wrong also I don't know what to do after this step?

 

[mighty2000]

Your Attempt at a Solution:

First, let's calculate (Vector a × Vector b) + (Vector a × Vector c):
(Vector a × Vector b) = (3, 4, 1) × (5, -2, 3) = (22, 12, -23)
(Vector a × Vector c) = (3, 4, 1) × (0, 1, -3) = (1, -9, 3)
Therefore, (Vector a × Vector b) + (Vector a × Vector c) = (22, 12, -23) + (1, -9, 3) = (23, 3, -20)

Now, let's calculate Vector a × (Vector b + Vector c):
(Vector b + Vector c) = (5, -2, 3) + (0, 1, -3) = (5, -1, 0)
Therefore, Vector a × (Vector b + Vector c) = (3, 4, 1) × (5, -1, 0) = (15, -4, 0)

As we can see, both calculations result in the same answer, so we can conclude that Vector a × (Vector b + Vector c) = (Vector a × Vector b) + (Vector a × Vector c). This is because the cross product is distributive over addition, as shown in the homework equation provided.