Question

What does a dot product represent?

Answer 1

Dot product:

1. The dot product of two vectors is a sum of the products of their corresponding horizontal and vertical components(in $xy-$plane).
2. The output is a scalar number.
3. The dot product of two vectors $\overrightarrow{A}$ and $\overrightarrow{B}$ is $\stackrel{\mathbf{\rightharpoonup }}{\mathbf{A}}\mathbf{·}\stackrel{\mathbf{\rightharpoonup }}{\mathbf{B}}\mathbf{=}\left|\stackrel{\rightharpoonup }{A}\right|\left|\stackrel{\rightharpoonup }{B}\right|\mathit{c}\mathit{o}\mathit{s}\mathit{\theta }$ , where $\theta$ is angle between them.
4. Another formula for the dot product of two vectors $\overrightarrow{A}=a_x\hat{i}+a_y\hat{\mathrm{j}}+a_z\hat{k}$ and $\overrightarrow{B}=b_x\hat{i}+b_y\hat{\mathrm{j}}+b_z\hat{k}$ is $\stackrel{\mathbf{\rightharpoonup }}{\mathbf{A}}\mathbf{·}\stackrel{\mathbf{\rightharpoonup }}{\mathbf{B}}\mathbf{=}{\mathit{a}}_{\mathbf{x}}{\mathit{b}}_{\mathbf{x}}\mathbf{+}{\mathit{a}}_{\mathbf{y}}{\mathit{b}}_{\mathbf{y}}\mathbf{+}{\mathit{a}}_{\mathbf{z}}{\mathit{b}}_{\mathbf{z}}$
5. Geometrically,it will also be equal to the product of ‘projection’ of magnitude of one vector on the other and magnitude of the 2nd vector.

Hence, meaning of dot product is explained.