Can a dot product be negative in case of length?

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SUMMARY

The dot product of two vectors A and B can indeed be negative when the angle between them is obtuse, such as 170°. In this case, the dot product yields a negative value in cm² due to the negative cosine of the angle. This indicates that the projection of vector A onto vector B occurs in the opposite direction of vector B. The magnitude of the vectors does not influence the sign of the dot product, as length is always non-negative.

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Let's say A and B are 2 vectors with length in cm and the angle between them is 170°.

Obviously, the dot product of A and B will give cm2 as unit but since the value of cos(170) is negative, will the dot product be negative (something)cm2?
 
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Yes, the dot product will be negative. The geometric implication is that when A is projected onto B, the projection will be in the opposite direction to B (and vice versa).

The length merely represents the magnitude of a vector, and not its direction. A length is always non-negative, but that doesn't stop the dot product of two vectors from being negative.
 
Last edited:
Curious3141 said:
Yes, the dot product will be negative. The geometric implication is that when A is projected onto B, the projection will be in the opposite direction to B (and vice versa).

The length merely represents the magnitude of a vector, and not its direction. A length is always non-negative, but that doesn't stop the dot product of two vectors from being negative.

Aha! Thanks.
 

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