Vector
Understanding Vectors
In mathematics and physics, a vector is a quantity that has both magnitude and direction. Vectors are commonly used to represent physical quantities such as velocity, force, and displacement.
Vector Representation
Vector can be represented in two ways:
- bold: $\mathbf{v}$
- With an arrow: $\vec{v}$
Vector consists of two froms: column from and row form.
- Column form: \(\mathbf{v} = \begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix}\)
- Row form: \(\mathbf{v} = \begin{pmatrix} v_1, v_2, v_3 \end{pmatrix}\)
Vector Operations
1. Vector Addition
The sum of two vectors $\mathbf{a}$ and $\mathbf{b}$ is given by: \(\mathbf{a} + \mathbf{b} = \begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix} + \begin{pmatrix} b_1 \\ b_2 \\ b_3 \end{pmatrix} = \begin{pmatrix} a_1 + b_1 \\ a_2 + b_2 \\ a_3 + b_3 \end{pmatrix}\)
2. Scalar Multiplication
Multiplying a vector $\mathbf{v}$ by a scalar $\lambda$ results in: \(\lambda \mathbf{v} = \lambda \begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix} = \begin{pmatrix} \lambda v_1 \\ \lambda v_2 \\ \lambda v_3 \end{pmatrix}\)
3. Dot Product (Scalar Product)
The dot product of two vectors $\mathbf{a}$ and $\mathbf{b}$ is defined as: \(\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3\) Alternatively, in terms of magnitudes and the angle $\theta$ between the vectors: \(\mathbf{a} \cdot \mathbf{b} = |\mathbf{a}| |\mathbf{b}| \cos \theta\) The dot product results in a scalar.
4. Cross Product (Vector Product)
The cross product of two vectors $\mathbf{a}$ and $\mathbf{b}$ in 3D is given by: \(\mathbf{a} \times \mathbf{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}\) This results in a new vector that is perpendicular to both $\mathbf{a}$ and $\mathbf{b}$, and its direction is given by the right-hand rule.
Magnitude of a Vector
The magnitude (or length) of a vector $\mathbf{v} = \begin{pmatrix} v_1 \ v_2 \ v_3 \end{pmatrix}$ is given by:
\[|\mathbf{v}| = \sqrt{v_1^2 + v_2^2 + v_3^2}\]For example, in 2D, if $\mathbf{v} = \begin{pmatrix} 3, \ 4 \end{pmatrix}$, the magnitude is: \(|\mathbf{v}| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = 5\)
Unit Vector
A unit vector is a vector that has a magnitude of 1 and points in the direction of a given vector.
The unit vector $\vec{\mathbf{v}}$ is calculated by:
\(\vec{\mathbf{v}} = \frac{\mathbf{v}}{|\mathbf{v}|}\)
For example, if $\mathbf{v} = \begin{pmatrix} 3 \ 4 \end{pmatrix}$ , the unit vector is:
\(\vec{\mathbf{v}} = \frac{1}{5} \begin{pmatrix} 3 \\ 4 \end{pmatrix} = \begin{pmatrix} \frac{3}{5} \\ \frac{4}{5} \end{pmatrix}\)
Application of Vectors
Vectors have a wide range of applications in various fields:
- Physics: Representing force, velocity, acceleration.
- Computer Graphics: Modeling directions and positions in 3D space.
- Engineering: Structural analysis, fluid dynamics.
- Machine Learning: Representing data points in high-dimensional space.
Conclusion
Vectors are a fundamental concept in mathematics and are used in many fields to describe quantities that have both direction and magnitude. By understanding vector operations such as addition, scalar multiplication, and dot/cross products, we can model and solve various real-world problems more effectively