Question

The dot product of force and velocity gives?

Answer 1

First we need to know what a dot product defines and how it is calculated. Then convert the velocity in terms of displacement and time. Now the dot product of force and displacement is denoted as work. After that we will get a relation between work and time which on further solving gives us the solution.

Complete step by step answer:
Dot product: Dot product is an algebraic operation that takes two equal length sequences of numbers and returns a single number.We need to tell what is the output of the dot product of force and velocity. We know force times velocity can also be written as force times displacement over time. Mathematically we can say,
$\overrightarrow F \cdot \overrightarrow v = \overrightarrow F \cdot \dfrac{{\overrightarrow d }}{t}$
Where, Force = $F$, Displemet = $d$ and Time taken = $t$.

We know force times displacement is equal to work.Now,
$\overrightarrow F \cdot \overrightarrow d = \overrightarrow W$
Now on putting in equation $\left( 1 \right)$ we will get,
$\overrightarrow F \cdot \dfrac{{\overrightarrow d }}{t} = \dfrac{W}{t}$
Work divided by time is equal to power.
Hence,
$\dfrac{W}{t} = P$
Where P = Power.

Therefore the dot product of force and velocity is power.

Additional information:
We can also solve this problem in another method by taking the vector component in a different direction that is along the x, y, and z direction. As force and velocity are vector quantity we can write it as,
$\overrightarrow {\text{F}} {\text{ = }}{{\text{F}}_{\text{x}}}{\text{i + }}{{\text{F}}_{\text{y}}}{\text{j + }}{{\text{F}}_{\text{z}}}{\text{k}}$
And $\overrightarrow {\text{v}} {\text{ = }}{{\text{v}}_{\text{x}}}{\text{i + }}{{\text{v}}_{\text{y}}}{\text{j + }}{{\text{v}}_{\text{z}}}{\text{k}}$

Taking the dot product we will get,
$\overrightarrow F \cdot \overrightarrow v = \left( {{{\text{F}}_{\text{x}}}{\text{i + }}{{\text{F}}_{\text{y}}}{\text{j + }}{{\text{F}}_{\text{z}}}{\text{k}}} \right) \cdot \left( {{{\text{v}}_{\text{x}}}{\text{i + }}{{\text{v}}_{\text{y}}}{\text{j + }}{{\text{v}}_{\text{z}}}{\text{k}}} \right)$
$\Rightarrow \overrightarrow F \cdot \overrightarrow v = \left( {{{\text{F}}_{\text{x}}}{{\text{v}}_{\text{x}}}{\text{ + }}{{\text{F}}_{\text{y}}}{{\text{v}}_{\text{y}}}{\text{ + }}{{\text{F}}_{\text{z}}}{{\text{v}}_{\text{z}}}} \right)$
Product of force and velocity gives power.
$\overrightarrow F \cdot \overrightarrow v = {{\text{P}}_x} + {P_y} + {P_z}$

Note: The dot product of two vectors always gives a scalar value and hence the power is also a scale quantity. Moreover remember that the scale value produced is closely related to the cosine of the angle between the two vectors.