Question

(a) For what values of the angle $\mathit{\theta }$between two vectors is their scalar product positive? (b) For what values of $\mathit{\theta }$ is their scalar product negative?

Short answer

(a)Thescalar product between two vectors is positive if the angle$\theta$ is $0°⩽\theta ⩽90°$.

(b) The scalar product between two vectors is negative if the angle $\theta$is $90°⩽\theta ⩽180°$.

Step-by-step solution

Given information

The angle between two vectors is, $\theta$.

Scalar product of vectors

Consider, there are two vectors $\overrightarrow{a}$ and $\overrightarrow{b}$, then the value oftheir scalar product is given by,

role="math" localid="1663752655773" $$\begin{gathered} \overrightarrow{\mathbf{a}}\mathbf{\times }\overrightarrow{\mathbf{b}}\mathbf{=}\mathit{a}\mathit{b}\mathit{c}\mathit{o}\mathit{s}\mathit{\theta } \\ \mathit{c}\mathit{o}\mathit{s}\mathit{\theta }\mathbf{=}\frac{\overrightarrow{\mathbf{a}}\mathbf{\times }\overrightarrow{\mathbf{b}}}{\mathbf{a}\mathbf{b}} \end{gathered}$$

Here, role="math" localid="1663752665666" $\mathit{\theta }$is the angle between two vectors.

The value of the scalar product of two vectors relies upon the magnitude of the two vectors and the cosine of the angle between two vectors.

Angle for a positive scalar product

(a)

The value of$\theta$ determines whether thescalar product is positive or negative.

The value of$\cos \theta$ is always positive for the interval $0°⩽\theta ⩽90°$.

Hence, the scalar product between two vectors is positive if the angle $\theta$is $0°⩽\theta ⩽90°$.

Angle for a negative scalar product

(b)

The value of$\cos \theta$ is always negative for the interval $90°⩽\theta ⩽180°$.

Hence, the scalar product between two vectors is negative if the angle is $90°⩽\theta ⩽180°$.