A vector is a quantity that has both magnitude and direction. It's often represented graphically as an arrow, where the length of the arrow indicates its magnitude, and the arrowhead points in its direction.

A vector can be denoted by a boldface letter (e.g., v), an arrow over a letter (e.g., \(\vec{v}\)), or by its initial and terminal points (e.g., \(\vec{AB}\)).

In a coordinate system, a vector can be broken down into its components along the axes.

• In 2D (a plane), a vector \(\vec{v}\) can be represented as \(\langle v_x, v_y \rangle\), where \(v_x\) is the horizontal component and \(v_y\) is the vertical component.
• In 3D (space), a vector \(\vec{v}\) can be represented as \(\langle v_x, v_y, v_z \rangle\).

The magnitude of a 2D vector \(\vec{v} = \langle v_x, v_y \rangle\) is calculated using the Pythagorean theorem: \(||\vec{v}|| = \sqrt{v_x^2 + v_y^2}\).

Vectors can be added, subtracted, and multiplied by scalars.

Vector Addition:

To add two vectors, \(\vec{u} = \langle u_x, u_y \rangle\) and \(\vec{v} = \langle v_x, v_y \rangle\), you add their corresponding components:

$$ \vec{u} + \vec{v} = \langle u_x + v_x, u_y + v_y \rangle $$

Graphically, this is done using the triangle method or parallelogram method.

Vector Subtraction:

Subtracting vectors is similar to adding the negative of a vector:

$$ \vec{u} - \vec{v} = \langle u_x - v_x, u_y - v_y \rangle $$

Scalar Multiplication:

Multiplying a vector \(\vec{v} = \langle v_x, v_y \rangle\) by a scalar (a real number) \(c\) scales its magnitude:

$$ c\vec{v} = \langle c \cdot v_x, c \cdot v_y \rangle $$

If \(c > 0\), the direction remains the same. If \(c < 0\), the direction reverses.

The dot product of two vectors \(\vec{u}\) and \(\vec{v}\) results in a scalar (a single number). It's defined as:

$$ \vec{u} \cdot \vec{v} = ||\vec{u}|| \cdot ||\vec{v}|| \cdot \cos(\theta) $$

where \(\theta\) is the angle between the two vectors. In terms of components, for \(\vec{u} = \langle u_x, u_y \rangle\) and \(\vec{v} = \langle v_x, v_y \rangle\):

$$ \vec{u} \cdot \vec{v} = u_x v_x + u_y v_y $$

The dot product is useful for finding the angle between vectors and determining if they are orthogonal (perpendicular, if \(\vec{u} \cdot \vec{v} = 0\)).

Example: If \(\vec{u} = \langle 1, 2 \rangle\) and \(\vec{v} = \langle 3, -1 \rangle\), then \(\vec{u} \cdot \vec{v} = (1)(3) + (2)(-1) = 3 - 2 = 1\).

The cross product of two vectors \(\vec{u}\) and \(\vec{v}\) results in a new vector that is perpendicular to both \(\vec{u}\) and \(\vec{v}\). This operation is only defined for 3D vectors.

For \(\vec{u} = \langle u_x, u_y, u_z \rangle\) and \(\vec{v} = \langle v_x, v_y, v_z \rangle\):

$$ \vec{u} \times \vec{v} = \langle u_y v_z - u_z v_y, u_z v_x - u_x v_z, u_x v_y - u_y v_x \rangle $$

The magnitude of the cross product is \(||\vec{u} \times \vec{v}|| = ||\vec{u}|| \cdot ||\vec{v}|| \cdot \sin(\theta)\), which represents the area of the parallelogram formed by the two vectors. Its direction is determined by the right-hand rule.

Vectors have widespread applications in various fields:

• Physics: Representing forces, velocities, accelerations, and displacements.
• Engineering: Structural analysis, fluid dynamics, and robotics.
• Computer Graphics: 3D modeling, animations, and game development.
• Navigation: Describing direction and distance for movement.
• Mathematics: Linear algebra, calculus (vector calculus), and geometry itself.