Use a specific example to prove that the cross product is also not associative. That is, use three specific vectors in 3-space to show that Vector a×(Vector b × Vector c) is not equal to (Vector a × Vector b) × Vector c.
Can you help me think of an example im having trouble doing so
Sure! Let's consider the three vectors in 3-space:
Vector a = <1, 2, 3>
Vector b = <4, 5, 6>
Vector c = <7, 8, 9>
Now, let's first calculate the cross product of Vector b and Vector c:
Vector b × Vector c = <4, 5, 6> × <7, 8, 9>
Using the formula for the cross product, we calculate the determinant:
<4, 5, 6> × <7, 8, 9> = <(5*9) - (6*8), (6*7) - (4*9), (4*8) - (5*7)>
= <-3, 6, -3>
Next, let's calculate the cross product of Vector a and the result we just obtained:
Vector a × (<-3, 6, -3>) = <1, 2, 3> × <-3, 6, -3>
Again, using the formula for the cross product, we calculate the determinant:
<1, 2, 3> × <-3, 6, -3> = <(2*(-3)) - (3*6), (3*(-3)) - (1*(-3)), (1*6) - (2*(-3))>
= <-12, -6, 12>
Now let's calculate the cross product of Vector a and Vector b:
<Vector a × Vector b> × Vector c = <-12, -6, 12> × <7, 8, 9>
Calculating the determinant:
<-12, -6, 12> × <7, 8, 9> = <(-6*9) - (12*8), (12*7) - (-12*9), (-12*8) - (-6*7)>
= <-150, 180, -6>
Comparing the two results, we have:
Vector a × (<-3, 6, -3>) = <-12, -6, 12>
<Vector a × Vector b> × Vector c = <-150, 180, -6>
As you can see, Vector a × (Vector b × Vector c) is not equal to (Vector a × Vector b) × Vector c.
Sure! Let's use a specific example to prove that the cross product is not associative.
Let's consider three vectors in 3-space:
Vector a = (1, 2, 3)
Vector b = (4, 5, 6)
Vector c = (7, 8, 9)
To show that Vector a×(Vector b × Vector c) is not equal to (Vector a × Vector b) × Vector c, we need to calculate both sides of the equation.
First, let's calculate Vector b × Vector c:
Vector b × Vector c = (4, 5, 6) × (7, 8, 9)
Using the formula for the cross product of two vectors, we get:
Vector b × Vector c = ((5 * 9) - (6 * 8), (6 * 7) - (4 * 9), (4 * 8) - (5 * 7))
= (-3, 6, -3)
Now, let's calculate Vector a × (Vector b × Vector c):
Vector a × (Vector b × Vector c) = (1, 2, 3) × (-3, 6, -3)
Again, using the formula for the cross product, we get:
Vector a × (Vector b × Vector c) = ((2 * -3) - (3 * 6), (3 * -3) - (1 * -3), (1 * 6) - (2 * -3))
= (-24, -6, 12)
Next, let's calculate (Vector a × Vector b) × Vector c:
(Vector a × Vector b) × Vector c = (1, 2, 3) × (4, 5, 6) × (7, 8, 9)
We need to first calculate Vector a × Vector b:
Vector a × Vector b = (1, 2, 3) × (4, 5, 6)
Using the cross product formula, we find:
Vector a × Vector b = ((2 * 6) - (3 * 5), (3 * 4) - (1 * 6), (1 * 5) - (2 * 4))
= (-7, 6, -3)
Now, let's substitute the above result into (Vector a × Vector b) × Vector c:
(Vector a × Vector b) × Vector c = (-7, 6, -3) × (7, 8, 9)
Applying the cross product formula one more time, we get:
(Vector a × Vector b) × Vector c = ((6 * 9) - (-3 * 8), (-3 * 7) - (-7 * 9), (-7 * 8) - (6 * 7))
= (75, 48, -98)
Comparing the two results, we see that Vector a×(Vector b × Vector c) = (-24, -6, 12) is not equal to (Vector a × Vector b) × Vector c = (75, 48, -98).
Therefore, we have proven that the cross product is not associative using this specific example.
come on. You can do a cross product, right? So, pick some vectors and do the math.
For example, let
u = <1,1,1>
v = <1,-1,1>
w = <2,1,3>
(uxv)xw = <2,0,-2>x<2,1,3> = <2,-10,2>
ux(vxw) = <1,1,1>x<2,-1,3> = <4,-1,-3>