Law of Vectors (Cross Product)

[Macleef]

Homework Statement

Prove:
(au) × v + (bu) × v = [(a + b)u] × v

Homework Equations

http://en.wikipedia.org/wiki/Cross_product

The Attempt at a Solution

u = ( x , y , z )
v = ( x₂, y₂,z₂)

LHS:
= (au) × v + (bu) × v
= [ ay z₂- y₂az , - (axz₂- x₂az) , axy₂- x₂ay ] + [ by z₂- y₂bz , - (bxz₂- x₂bz) , bxy₂- x₂by ]RHS:
= [(a + b)u] × v
= [ (a + b)x , (a + b)y , (a + b)z ] × ( x₂, y₂,z₂)
= [ (a + b)y z₂- y₂(a + b)z , -((a + b)x z₂- x₂(a + b)z) , (a + b)x y₂- x₂(a + b)y]

This is how far I got to prove left side equals right side. . .now I don't know what to do.

 

[PingPong]

Homework Statement

Prove:
(au) × v + (bu) × v = [(a + b)u] × v

Homework Equations

http://en.wikipedia.org/wiki/Cross_product

The Attempt at a Solution

u = ( x , y , z )
v = ( x₂, y₂,z₂)

LHS:
= (au) × v + (bu) × v
= [ ay z₂- y₂az , - (axz₂- x₂az) , axy₂- x₂ay ] + [ by z₂- y₂bz , - (bxz₂- x₂bz) , bxy₂- x₂by ]


RHS:
= [(a + b)u] × v
= [ (a + b)x , (a + b)y , (a + b)z ] × ( x₂, y₂,z₂)
= [ (a + b)y z₂- y₂(a + b)z , -((a + b)x z₂- x₂(a + b)z) , (a + b)x y₂- x₂(a + b)y]

This is how far I got to prove left side equals right side. . .now I don't know what to do.

Remember that vectors add component-wise (that is, the x-components add when adding two vectors), and that two vectors are equal if their components are equal.

So, you're almost there! Simplify the left hand side a bit more by combining it into one vector instead of the sum of two vectors and compare the components!