Question

What is known about \(\theta,\) the angle between two nonzero vectors \(\mathbf{u}\) and \(\mathbf{v},\) if (a) \(\mathbf{u} \cdot \mathbf{v}=0\) ? (b) \(\mathbf{u} \cdot \mathbf{v}>0 ?\) (c) \(\mathbf{u} \cdot \mathbf{v}<0 ?\)

Short answer

(a) The vectors are perpendicular; (b) The angle between vectors is acute; (c) The angle between vectors is obtuse.

Step-by-step solution

Interpretation of dot product

The dot product of two vectors \(\mathbf{u}\) and \(\mathbf{v}\) is given by \(\mathbf{u} \cdot \mathbf{v} = ||\mathbf{u}|| ||\mathbf{v}|| \cos(\theta)\), where ||\mathbf{u}|| and ||\mathbf{v}|| are the magnitudes of the vectors, and \(\theta\) is the angle between them. Because both vectors are non-zero, their magnitudes are also non-zero. Hence, the dot product equals zero or is positive/negative depending on the value of \(\cos(\theta)\).

Case (a): Dot product equals zero

If \(\mathbf{u} \cdot \mathbf{v}=0\), then according to the formula from step 1, the \(cos(\theta) = 0\). The only possible value for \(\theta\) that makes \(cos(\theta) = 0\) in the interval between 0 and 180 degrees is \(\theta = 90\) degrees. Therefore, in this case, vectors \(\mathbf{u}\) and \(\mathbf{v}\) are perpendicular.

Case (b): Dot product is positive

If \(\mathbf{u} \cdot \mathbf{v}>0\), then \(cos(\theta) > 0\). Cosine is positive for angles in the interval between 0 and 90 degrees. Therefore, in this case, \(\theta\) is an acute angle.

Case (c): Dot product is negative

If \(\mathbf{u} \cdot \mathbf{v}<0\), then \(cos(\theta) < 0\). Cosine is negative for angles in the interval between 90 and 180 degrees. Therefore, in this case, \(\theta\) is an obtuse angle.