Question

How do I find the dot product of two three – dimensional vectors?

Answer 1

Hint: The vectors are defined as an object containing both magnitude and direction. Vector describes the movement of an object from one point to another. Vector math can be geometrically posterized by the directed line segment.

Complete step by step solution:
The dot product of two vectors always results in scalar quantity, i.e. it has only magnitude and no direction. It is represented by a dot (.) in between two vectors.
a dot b = a. b
A three dimensional is a vector which has its coordinate in all three axes, we also know that the axis in vectors is represented as \[\widehat i,\widehat j,\widehat k\]. If we dot the same unit vector we will get \[1\] and so on.
In order to find the dot product let us assume two three-dimensional vectors as,
\[{V_1} = a\widehat i + b\widehat j + c\widehat k{\text{ and }}{{\text{V}}_2} = x\widehat i{\text{ + y}}\widehat j{\text{ + z}}\widehat k\]
So, the dot product will be,
\[{V_1}.{{\text{V}}_2} = (a\widehat i + b\widehat j + c\widehat k).(x\widehat i{\text{ + y}}\widehat j{\text{ + z}}\widehat k)\]
=\[(a..x) + (b.{\text{y)}} + (c.{\text{z)}}\]
We have to multiply the coefficient of the same unit vectors only.
We can also find the dot product if we know the angle in between the two vectors.
$\overrightarrow a .\overrightarrow b = \left| a \right|\left| b \right|\cos \theta $ where $\left| a \right|{\text{ and }}\left| b \right|$ are the magnitude of two vectors.

Note:
Dot product of any two vectors will be a scalar quantity only, while the cross product will always yield us another vector.
Multiplication of vectors is of two types: one is the scalar product i.e. the dot product which we saw above, another is the vector product which is also known as cross product of two vectors.