1. Meaning of the Dot Product
The dot product (also called the scalar product) is a way to multiply two vectors and get a real number. This number tells how much the two vectors point in the same direction.
If two vectors point in similar directions, their dot product is positive. If they are perpendicular, the dot product is zero. If they point in opposite directions, the dot product is negative.
2. Algebraic Formula of Dot Product
When vectors are written in component form, the dot product is found by multiplying corresponding components and adding the results.
For 2D vectors:
\[ \langle a_1, b_1 \rangle \cdot \langle a_2, b_2 \rangle = a_1a_2 + b_1b_2 \]
For 3D vectors:
\[ \langle a_1, b_1, c_1 \rangle \cdot \langle a_2, b_2, c_2 \rangle = a_1a_2 + b_1b_2 + c_1c_2 \]
2.1. Example
Let:
\[ \vec{u} = \langle 2, -1, 3 \rangle \quad \text{and} \quad \vec{v} = \langle 1, 4, -2 \rangle \]
Then:
\[ \vec{u} \cdot \vec{v} = 2(1) + (-1)(4) + 3(-2) = 2 - 4 - 6 = -8 \]
3. Geometric Formula of Dot Product
The dot product also has a geometric meaning. It relates the magnitudes of the vectors and the angle between them.
\[ \vec{u} \cdot \vec{v} = |\vec{u}| \, |\vec{v}| \, \cos \theta \]
Here, \(\theta\) is the angle between the vectors.
This formula helps decide whether two vectors are perpendicular, pointing the same way, or opposite.
3.1. Perpendicular Case
If two vectors are perpendicular, the angle between them is \(90^\circ\). Since:
\[ \cos 90^\circ = 0 \]
The dot product becomes:
\[ \vec{u} \cdot \vec{v} = 0 \]
This gives a quick test for perpendicularity.
3.2. Parallel and Opposite Case
If vectors point in the same direction, \(\theta = 0^\circ\) and:
\[ \cos 0 = 1 \]
If they point in opposite directions, \(\theta = 180^\circ\) and:
\[ \cos 180^\circ = -1 \]
4. Using the Dot Product to Find the Angle
You can rearrange the geometric formula to calculate the angle between two vectors:
\[ \cos \theta = \dfrac{\vec{u} \cdot \vec{v}}{|\vec{u}| \, |\vec{v}|} \]
4.1. Example
Let:
\[ \vec{u} = \langle 3, 4 \rangle, \quad \vec{v} = \langle 4, 3 \rangle \]
Dot product:
\[ \vec{u} \cdot \vec{v} = 3(4) + 4(3) = 24 \]
Magnitudes:
\[ |\vec{u}| = 5, \quad |\vec{v}| = 5 \]
So:
\[ \cos \theta = \dfrac{24}{25} \]
The angle is small, meaning the vectors point almost in the same direction.
5. Projection of One Vector on Another
The dot product helps find how much of one vector lies along another. This is called the projection.
The projection of \(\vec{u}\) on \(\vec{v}\) is:
\[ \text{proj}_{\vec{v}}(\vec{u}) = \dfrac{\vec{u} \cdot \vec{v}}{|\vec{v}|^2} \vec{v} \]
5.1. Example
For \(\vec{u} = \langle 2, 1 \rangle\) and \(\vec{v} = \langle 3, 0 \rangle\):
Dot product:
\[ \vec{u} \cdot \vec{v} = 6 \]
Then:
\[ \text{proj}_{\vec{v}}(\vec{u}) = \dfrac{6}{9} \langle 3, 0 \rangle = \langle 2, 0 \rangle \]
6. Important Properties of the Dot Product
- Dot product is commutative: \( \vec{u} \cdot \vec{v} = \vec{v} \cdot \vec{u} \)
- It distributes over addition: \( \vec{u} \cdot (\vec{v} + \vec{w}) = \vec{u} \cdot \vec{v} + \vec{u} \cdot \vec{w} \)
- Dot product of a vector with itself gives squared magnitude: \( \vec{v} \cdot \vec{v} = |\vec{v}|^2 \)
- Zero vector has zero dot product with all vectors.