# Vectors

Vectors are mathematical objects that are commonly used in various fields of science, engineering, and mathematics. They are used to represent physical quantities such as force, velocity, acceleration, and displacement that have both magnitude and direction.

A vector is usually represented by an arrow with a length and a direction. The length of the arrow represents the magnitude of the vector, while the direction of the arrow represents the direction of the vector. The magnitude of a vector is a scalar value and is denoted by \( \vec{|v|} \).

Vectors can be denoted by naming their start and end points, where the start point is the tail of the vector and the end point is the head of the vector. For example, the vector from point \(A\) to point \(B\) can be denoted as \( \overrightarrow{AB} \) .

**Vectors have a number of properties that are important in mathematics and physics. Some of the key properties of vectors include:**

**Magnitude:** Vectors have a magnitude or length, which is a non-negative scalar that represents the size of the vector.

**Direction:** Vectors have a direction, which can be specified using angles or other directional notations. The direction of a vector is defined by the angle between the vector and a fixed reference direction.

**Addition:** Vectors can be added together using the parallelogram law or the triangle law of vector addition. This involves adding the corresponding components of each vector to obtain the resulting vector.

**Scalar multiplication:** Vectors can be multiplied by scalars, which changes the magnitude and/or direction of the vector. Scalar multiplication involves multiplying each component of the vector by a scalar.

**Dot product:** Vectors can be multiplied together using the dot product or scalar product. The dot product of two vectors is a scalar that represents the product of their magnitudes and the cosine of the angle between them.

**Cross product:** Vectors can also be multiplied using the cross product or vector product. The cross product of two vectors is a vector that is perpendicular to both input vectors and has a magnitude equal to the product of their magnitudes times the sine of the angle between them.

**Zero vector:** There is a unique vector called the zero vector, denoted by \( \vec{0} \), which has a magnitude of \(0\) and no direction. It can be thought of as the vector that goes from a point to itself, or equivalently, as the difference between any two equal vectors.

For example, \( \overrightarrow{OA} - \overrightarrow{OA} = 0 \).

**Unit vector:** A unit vector is a vector with a magnitude of 1. Any non-zero vector can be divided by its magnitude to obtain a unit vector in the same direction.

**Collinear vectors:** Vectors are collinear if they lie on the same line or are parallel. In other words, they have the same or opposite direction. Collinear vectors can be written as scalar multiples of each other. If two vectors \( \vec{v} \) and \( \vec{w} \) are collinear, then there exists a scalar \(k\) such that \( \vec{v} = k \vec{w} \) or \( \vec{w} = k \vec{v} \). This means that one vector is a scalar multiple of the other, and they point in the same or opposite directions.

**Orthogonal vectors:** Two vectors are orthogonal if their dot product is zero. Orthogonal vectors are also called perpendicular vectors, and they form a 90-degree angle between them.

**Basis vectors:** A set of basis vectors is a set of linearly independent vectors that can be used to represent any other vector in a space. The most common basis vectors are the standard unit vectors in three-dimensional space, which are denoted by \(\hat{i}\) , \( \hat{j} \) and \( \hat{k} \).

**Linear independence:** A set of vectors is linearly independent if none of the vectors in the set can be expressed as a linear combination of the others. If a set of vectors is linearly independent, then it can be used as a basis for a vector space.

**Span:** The span of a set of vectors is the set of all linear combinations of those vectors. The span of a set of vectors is a subspace of the vector space containing those vectors.

**Projection:** The projection of one vector onto another is the component of the first vector that lies in the direction of the second vector. The projection of vector \( \vec{u} \) onto vector \(\vec{v}\) is given by $$ \text{proj}_{\vec{v}} ( \vec{u} ) = \frac{\vec{u} \cdot v}{||\vec{v}||^2} \vec{v} $$

**Component:** The component of one vector along another is the part of the first vector that lies in the direction of the second vector. The component of vector \( \vec{u} \) along vector \( \vec{v} \) is given by $$ \text{comp}_{\vec{v}} (\vec{u}) = \frac{\vec{u} \cdot v}{||\vec{v}||^2} cos \theta $$ where \(\theta \) is the angle between \(\vec{u} \) and \(\vec{v} \).

**Parallel transport:** Parallel transport is a way of moving vectors along a curve in a way that preserves their direction. Parallel transport is used in differential geometry and other fields to study the curvature of curves and surfaces.

**Covariance and contravariance:** In mathematics and physics, vectors are often classified as covariant or contravariant based on how their components transform under coordinate transformations. Covariant vectors have components that transform in the same way as the coordinates, while contravariant vectors have components that transform in the opposite way. The concept of covariance and contravariance is used extensively in tensor calculus and other areas of mathematics and physics.

These are just a few of the many properties of vectors.