Are the two vectors (6,6,6,0,0,0) and (0,0,0,6,6,6) perpendicular?

Question

BlackTom AI · Accepted Answer

To assess whether two vectors are perpendicular, we use the dot product: if it equals zero, the vectors are orthogonal. In this case, the vectors are v = (6,6,6,0,0,0) and w = (0,0,0,6,6,6).
Compute the dot product: v · w = 6*0 + 6*0 + 6*0 + 0*6 + 0*6 + 0*6 = 0.
Because the sum of the pairwise products is zero, the vectors are perpendicular.
If there were any alternative statements like 'they have a nonzero dot product' or 'they are parallel,' those would be incorrect, since a nonzero dot product indicates an angle not equal to 90 degrees, and parallel vectors would have a dot product equal to the product of their magnitudes with the cosine of the angle (which would be either the same direction or opposite), not zero in this setup.
Another way to see this is that the nonzero components of the first vector lie in the first three coordinates, while the nonzero components of the second vector lie in the last three coordinates; since these coordinate positions do not overlap, the cross-term products are all zeros, yielding a zero dot product.
Thus, the perpendicularity follows directly from the zero dot product criterion for orthogonality.

Are the two vectors (6,6,6,0,0,0) and (0,0,0,6,6,6) perpendicular?单项选择题

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