Can a dot product be negative in case of length?

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The dot product of two vectors can indeed be negative, even though the lengths of the vectors are always non-negative. In the case of vectors A and B with an angle of 170° between them, the dot product will yield a negative value due to the negative cosine of that angle. This negative result indicates that the projection of one vector onto the other is in the opposite direction. The discussion emphasizes that while vector lengths are non-negative, the directionality of vectors allows for a negative dot product. Understanding this concept is crucial in vector analysis.
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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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