Law of Vectors (Cross Product)

In summary, to prove (au) × v + (bu) × v = [(a + b)u] × v, first simplify the left hand side by combining it into one vector. Then, compare the components of the left and right hand sides to show that they are equal.
  • #1
Macleef
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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.
 
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  • #2
Macleef said:

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!
 

Related to Law of Vectors (Cross Product)

1. What is the cross product of two vectors?

The cross product of two vectors is a vector that is perpendicular to both of the original vectors and has a magnitude equal to the product of their magnitudes multiplied by the sine of the angle between them.

2. How is the cross product calculated?

The cross product of two vectors, a and b, is calculated using the following formula: a x b = (axby - aybx)i + (aybz - azby)j + (azbx - axbz)k, where i, j, and k are the unit vectors in the x, y, and z directions, respectively.

3. What is the significance of the direction of the cross product?

The direction of the cross product is significant because it tells us the direction in which the resulting vector is pointing. The right-hand rule is commonly used to determine the direction of the cross product, where the fingers of the right hand curl in the direction of the first vector and then point in the direction of the second vector to determine the direction of the resulting vector.

4. Can the cross product of two vectors be equal to zero?

Yes, the cross product of two vectors can be equal to zero if the vectors are parallel or if one of the vectors has a magnitude of zero.

5. What are some real-life applications of the cross product?

The cross product has many real-life applications, including calculating the torque on a rotating object, determining the direction of a magnetic field, and predicting the direction of the resulting force when two objects collide in a physics problem.

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