KAHIBARO
Table of Contents
Meaning and Definition
The dot product is a way to combine two vectors and get a single number. Because the result is a number, not another vector, the dot product is a scalar operation.
If two vectors $\vec{A}$ and $\vec{B}$ make an angle $\theta$ with each other, their dot product is defined by
$$
\vec{A} \cdot \vec{B} = AB \cos \theta
$$
Here, $A = |\vec{A}|$ and $B = |\vec{B}|$ are the magnitudes of the vectors.
This definition shows that the dot product depends on both the lengths of the vectors and the angle between them.
Important formula:
$$
\vec{A} \cdot \vec{B} = AB \cos \theta
$$
The dot product gives a scalar, not a vector.
Geometric Interpretation
The dot product tells us how much one vector points in the same direction as another. If two vectors point mostly in the same direction, the dot product is positive. If they are perpendicular, the dot product is zero. If they point mostly in opposite directions, the dot product is negative.
You can think of it as multiplying the magnitude of one vector by the component of the other along its direction.
If we project $\vec{B}$ onto $\vec{A}$, then
$$
\vec{A} \cdot \vec{B} = A \left(B \cos \theta\right)
$$
and $B \cos \theta$ is the part of $\vec{B}$ along $\vec{A}$.
Dot Product in Components
In Cartesian coordinates, vectors can be written in terms of components. If
$$
\vec{A} = (A_x, A_y, A_z), \qquad \vec{B} = (B_x, B_y, B_z)
$$
then their dot product is
$$
\vec{A} \cdot \vec{B} = A_xB_x + A_yB_y + A_zB_z
$$
In two dimensions, this becomes
$$
\vec{A} \cdot \vec{B} = A_xB_x + A_yB_y
$$
This component formula is often the easiest way to calculate the dot product.
Component form of the dot product:
$$
\vec{A} \cdot \vec{B} = A_xB_x + A_yB_y + A_zB_z
$$
Special Cases
Some important cases follow directly from the definition.
If the vectors are parallel and point in the same direction, then $\theta = 0$ and $\cos 0 = 1$, so
$$
\vec{A} \cdot \vec{B} = AB
$$
If the vectors are perpendicular, then $\theta = 90^\circ$ and $\cos 90^\circ = 0$, so
$$
\vec{A} \cdot \vec{B} = 0
$$
If the vectors point in opposite directions, then $\theta = 180^\circ$ and $\cos 180^\circ = -1$, so
$$
\vec{A} \cdot \vec{B} = -AB
$$
This makes the dot product very useful for checking whether two vectors are perpendicular.
Orthogonality rule:
If
$$
\vec{A} \cdot \vec{B} = 0
$$
and neither vector is zero, then the vectors are perpendicular.
Dot Product of a Vector with Itself
When a vector is dotted with itself, the angle is zero, so
$$
\vec{A} \cdot \vec{A} = A^2
$$
Using components,
$$
\vec{A} \cdot \vec{A} = A_x^2 + A_y^2 + A_z^2
$$
Therefore the magnitude of a vector can be found from
$$
|\vec{A}| = \sqrt{\vec{A} \cdot \vec{A}}
$$
This is an important link between the dot product and vector length.
Unit Vector Results
For the standard Cartesian unit vectors $\hat{i}$, $\hat{j}$, and $\hat{k}$, the dot products are
$$
\hat{i}\cdot\hat{i} = 1, \qquad \hat{j}\cdot\hat{j} = 1, \qquad \hat{k}\cdot\hat{k} = 1
$$
and
$$
\hat{i}\cdot\hat{j} = 0, \qquad \hat{i}\cdot\hat{k} = 0, \qquad \hat{j}\cdot\hat{k} = 0
$$
These relations come from the fact that each unit vector has magnitude 1 and different coordinate directions are perpendicular.
| Dot product | Value |
|---|---|
| $\hat{i}\cdot\hat{i}$ | $1$ |
| $\hat{j}\cdot\hat{j}$ | $1$ |
| $\hat{k}\cdot\hat{k}$ | $1$ |
| $\hat{i}\cdot\hat{j}$ | $0$ |
| $\hat{i}\cdot\hat{k}$ | $0$ |
| $\hat{j}\cdot\hat{k}$ | $0$ |
Finding the Angle Between Two Vectors
If the vectors are known, the dot product can be used to find the angle between them. Starting from
$$
\vec{A} \cdot \vec{B} = AB \cos \theta
$$
we solve for $\cos \theta$:
$$
\cos \theta = \frac{\vec{A} \cdot \vec{B}}{AB}
$$
So the angle is
$$
\theta = \cos^{-1}\left(\frac{\vec{A} \cdot \vec{B}}{AB}\right)
$$
This is useful in geometry and physics whenever direction matters.
Angle formula:
$$
\theta = \cos^{-1}\left(\frac{\vec{A} \cdot \vec{B}}{|\vec{A}|\,|\vec{B}|}\right)
$$
This formula works only if both vectors are nonzero.
Basic Properties
The dot product has several simple algebraic properties:
$$
\vec{A}\cdot\vec{B} = \vec{B}\cdot\vec{A}
$$
so it is commutative.
Also,
$$
\vec{A}\cdot(\vec{B}+\vec{C}) = \vec{A}\cdot\vec{B} + \vec{A}\cdot\vec{C}
$$
so it distributes over vector addition.
And for a scalar $c$,
$$
(c\vec{A})\cdot\vec{B} = c(\vec{A}\cdot\vec{B})
$$
These properties make dot product calculations easier.
Simple Example
Let
$$
\vec{A} = (2,3), \qquad \vec{B} = (4,1)
$$
Then
$$
\vec{A}\cdot\vec{B} = (2)(4) + (3)(1) = 8 + 3 = 11
$$
So the dot product is 11.
Now find the magnitudes:
$$
|\vec{A}| = \sqrt{2^2+3^2} = \sqrt{13}
$$
$$
|\vec{B}| = \sqrt{4^2+1^2} = \sqrt{17}
$$
Then
$$
\cos\theta = \frac{11}{\sqrt{13}\sqrt{17}}
$$
which gives the angle between the vectors.
Physical Meaning in Introductory Physics
In physics, the dot product appears whenever only the part of one vector in the direction of another matters. A force may act at an angle, but only its component along the displacement contributes to certain calculations. A velocity may have many components, but sometimes only the component in a chosen direction matters.
So the dot product is a compact way to express directional alignment between two vectors.
Summary
The dot product combines two vectors into a scalar. It is defined by
$$
\vec{A}\cdot\vec{B} = AB\cos\theta
$$
and in components by
$$
\vec{A}\cdot\vec{B} = A_xB_x + A_yB_y + A_zB_z
$$
It is positive for vectors pointing mostly in the same direction, zero for perpendicular vectors, and negative for vectors pointing mostly in opposite directions. It is also useful for finding magnitudes and angles between vectors.
Key ideas to remember:
$$
\vec{A}\cdot\vec{B} = AB\cos\theta
$$
$$
\vec{A}\cdot\vec{B} = A_xB_x + A_yB_y + A_zB_z
$$
$$
\vec{A}\cdot\vec{A} = |\vec{A}|^2
$$
If $\vec{A}\cdot\vec{B}=0$ for nonzero vectors, then the vectors are perpendicular.