Determine whether the vectors (3, 4, 5) and (4, -8, 12) are orthogonal, and explain your reasoning.
user
Sure, I'd be happy to help with that.

Two vectors in 3D space are orthogonal, or perpendicular, if their dot product is equal to zero. The dot product is calculated by multiplying the corresponding components of the two vectors and then adding them up.

So, let's calculate the dot product between (3, 4, 5) and (4, -8, 12):

Dot product = (3 * 4) + (4 * -8) + (5 * 12)
              = 12 - 32 + 60
              = 16

Since this is not equal to zero, these vectors are not orthogonal. They are actually linearly dependent because not all their components are zero. For example, you can express the second vector as a linear combination of the first vector:

(4, -8, 12) = (3, 4, 5) * 4 + (-1, -4, -7) * 1
                 = 4 * (3, 4, 5) - (1, -4, -7)

So, the vectors are not orthogonal.
assistant
Determine whether the vectors (3, 4, 5) and (4, -8, 12) are orthogonal, and explain your reasoning.
user
Two vectors are orthogonal if their dot product is equal to zero. Let's calculate the dot product of the given vectors (3, 4, 5) and (4, -8, 12):

Dot product = (3 * 4) + (4 * -8) + (5 * 12) = 12 - 32 + 60 = 40

Since the dot product is not equal to zero, the vectors (3, 4, 5) and (4, -8, 12) are not orthogonal.
assistant