Question

What if the dot product is negative?

Answer 1

Dot Product:

The dot product of two vectors is the sum of the products of their corresponding components. It is the product of their magnitudes multiplied by the cosine of the angle between them.

Formulas: The scalar product of two vectors $\overrightarrow{A}$ and $\overrightarrow{B}$ of magnitude $\left|\overrightarrow{A}\right|$ and $\left|\overrightarrow{A}\right|$ is $\overrightarrow{\mathbf{A}}\mathbf{.}\overrightarrow{\mathbf{B}}\mathbf{=}\left|\overrightarrow{A}\right|\left|\overrightarrow{B}\right|\mathbf{cos\theta }$

Or dot product of vectors $\overrightarrow{A}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}$ and $\overrightarrow{B}=b_1\hat{i}+b_2\hat{j}+b_3\hat{k}$ is $\overrightarrow{\mathbf{A}}\mathbf{.}\overrightarrow{\mathbf{B}}\mathbf{=}{\mathbf{a}}_{\mathbf{1}}{\mathbf{b}}_{\mathbf{1}}\mathbf{+}{\mathbf{a}}_{\mathbf{2}}{\mathbf{b}}_{\mathbf{2}}\mathbf{+}{\mathbf{a}}_{\mathbf{3}}{\mathbf{b}}_{\mathbf{3}}$

If the dot product of two vectors is negative, then the angle between the two vectors is greater than $90°$

Example: Let there be two vectors $\overrightarrow{A}=\hat{i}+2\hat{j}$ and $\overrightarrow{B}=3\hat{i}-5\hat{j}$ , then the dot product of the vectors is

$\overrightarrow{A}.\overrightarrow{B}=\left(1\right)\left(3\right)+\left(2\right)\left(-5\right)$

$\Rightarrow$$\overrightarrow{A}.\overrightarrow{B}=3-10$

$\Rightarrow$$\overrightarrow{A}.\overrightarrow{B}=-7$

Here $\left|\overrightarrow{A}\right|$,$\left|\overrightarrow{B}\right|$ are positive so $\mathrm{cos\theta }$ will be negative.

Hence, if dot product is negative then angle between the two vectors is greater than $\mathbf{90}\mathbf{°}$.