Scalar product
Scalar Product (Dot Product)
The scalar product, also known as the dot product, is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. This operation is widely used in physics, engineering, and mathematics, particularly in the context of vector algebra.
Definition
Given two vectors $\vec{a}$ and $\vec{b}$ in Euclidean space, their scalar product $\vec{a} \cdot \vec{b}$ is defined as:
$$ \vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos(\theta) $$
where:
- $|\vec{a}|$ is the magnitude (or length) of vector $\vec{a}$
- $|\vec{b}|$ is the magnitude of vector $\vec{b}$
- $\theta$ is the angle between $\vec{a}$ and $\vec{b}$
In Cartesian coordinates, for vectors $\vec{a} = (a_1, a_2, a_3)$ and $\vec{b} = (b_1, b_2, b_3)$, the scalar product can be computed as:
$$ \vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 + a_3b_3 $$
Properties of Scalar Product
The scalar product has several important properties:
- Commutative: $\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a}$
- Distributive: $\vec{a} \cdot (\vec{b} + \vec{c}) = \vec{a} \cdot \vec{b} + \vec{a} \cdot \vec{c}$
- Associative with scalar multiplication: $(k\vec{a}) \cdot \vec{b} = k(\vec{a} \cdot \vec{b}) = \vec{a} \cdot (k\vec{b})$ where $k$ is a scalar
- Orthogonality: If $\vec{a} \cdot \vec{b} = 0$ and $\vec{a} \neq \vec{0}$ and $\vec{b} \neq \vec{0}$, then $\vec{a}$ and $\vec{b}$ are orthogonal (perpendicular to each other)
Applications
- Projection: The scalar product can be used to find the projection of one vector onto another.
- Work: In physics, the work done by a force $\vec{F}$ moving an object through a displacement $\vec{d}$ is calculated as $\vec{F} \cdot \vec{d}$.
- Angle between vectors: The scalar product can be used to find the cosine of the angle between two vectors.
Examples
Let's consider two vectors $\vec{a} = (3, -2, 5)$ and $\vec{b} = (-1, 4, -2)$.
- Calculate their scalar product:
$$ \vec{a} \cdot \vec{b} = (3)(-1) + (-2)(4) + (5)(-2) = -3 - 8 - 10 = -21 $$
- Determine if they are orthogonal:
Since $\vec{a} \cdot \vec{b} \neq 0$, the vectors are not orthogonal.
- Find the angle between $\vec{a}$ and $\vec{b}$:
First, we find the magnitudes of $\vec{a}$ and $\vec{b}$:
$$ |\vec{a}| = \sqrt{3^2 + (-2)^2 + 5^2} = \sqrt{9 + 4 + 25} = \sqrt{38} $$
$$ |\vec{b}| = \sqrt{(-1)^2 + 4^2 + (-2)^2} = \sqrt{1 + 16 + 4} = \sqrt{21} $$
Now, we can find the cosine of the angle using the scalar product:
$$ \cos(\theta) = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}| |\vec{b}|} = \frac{-21}{\sqrt{38} \sqrt{21}} $$
The angle $\theta$ can then be found by taking the inverse cosine (arccos) of the above expression.
Comparison Table
| Property | Scalar Product | Vector Product |
|---|---|---|
| Result | Scalar (number) | Vector |
| Symbol | $\cdot$ | $\times$ |
| Commutative | Yes | No |
| Distributive | Yes | Yes |
| Orthogonality | $\vec{a} \cdot \vec{b} = 0$ implies orthogonality | $\vec{a} \times \vec{b} = \vec{0}$ implies parallelism or one is the zero vector |
| Application | Projection, work, angle between vectors | Cross product, torque, area of parallelogram |
In conclusion, the scalar product is a fundamental operation in vector algebra with a wide range of applications in various fields. Understanding its properties, how to calculate it, and its implications is essential for students and professionals working with vectors.