Find the cross product a × b.a = 8, 0, −3,    b = 0, 8, 0⟨24,0,64⟩ Verify that it is orthogonal to both a and b.

To find the cross product of two vectors, a and b, we can use the formula:

a × b = (a2b3 - a3b2, a3b1 - a1b3, a1b2 - a2b1)

Given that a = 8, 0, -3 and b = 0, 8, 0, we can substitute these values into the formula:

a × b = (0*(-3) - (-3)8, (-3)0 - 88, 88 - 0*0) = (0 + 24, 0 - 64, 64 - 0) = (24, -64, 64)

Therefore, the cross product of a and b is ⟨24, -64, 64⟩.

To verify that this cross product is orthogonal to both a and b, we can take the dot product of the cross product with each vector and check if the result is zero.

Dot product of a × b and a: (248) + (-640) + (64*(-3)) = 192 + 0 - 192 = 0

Dot product of a × b and b: (240) + (-648) + (64*0) = 0 - 512 + 0 = -512 + 0 = 0

Since the dot product of the cross product with both a and b is zero, we can conclude that the cross product ⟨24, -64, 64⟩ is orthogonal to both vectors a and b.