Vectors: Magnitude, Direction, and Operations

Vectors in Trigonometry

Introduction to Vectors

Vectors are mathematical objects that have both magnitude (length) and direction. They are commonly used in physics and mathematics to represent quantities such as displacement, velocity, and force.

• Magnitude: The length of the vector, often denoted as $|\vec{v}|$.

• Direction: The orientation of the vector in space, usually measured as an angle from a reference axis.

Vector Notation and Representation

Vectors can be represented graphically as arrows on a coordinate plane. The arrow points in the direction of the vector, and its length corresponds to the magnitude.

• Notation: Vectors are typically written in bold (e.g., v) or with an arrow above (e.g., $\vec{v}$).

• Component Form: A vector in the plane can be written as $\vec{v} = \langle v_x, v_y \rangle$, where $v_x$ and $v_y$ are the horizontal and vertical components, respectively.

Vector Addition and Subtraction

Vectors can be added or subtracted graphically or algebraically:

• Graphical Addition: Place the tail of the second vector at the head of the first vector. The resultant vector is drawn from the tail of the first to the head of the second.

• Algebraic Addition: Add corresponding components: $\vec{u} + \vec{v} = \langle u_x + v_x, u_y + v_y \rangle$

• Subtraction: Subtract corresponding components: $\vec{u} - \vec{v} = \langle u_x - v_x, u_y - v_y \rangle$

Example: Vector Operations

Given vectors $\vec{u}$ and $\vec{v}$, find $\vec{u} + \vec{v}$ and $\vec{u} - \vec{v}$.

• Suppose $\vec{u} = \langle 2, 3 \rangle$ and $\vec{v} = \langle 1, -1 \rangle$.

• Addition: $\vec{u} + \vec{v} = \langle 2+1, 3+(-1) \rangle = \langle 3, 2 \rangle$

• Subtraction: $\vec{u} - \vec{v} = \langle 2-1, 3-(-1) \rangle = \langle 1, 4 \rangle$

Magnitude of a Vector

The magnitude (length) of a vector $\vec{v} = \langle v_x, v_y \rangle$ is calculated using the Pythagorean theorem:

$|\vec{v}| = \sqrt{v_x^2 + v_y^2}$

Direction of a Vector

The direction (angle $\theta$) of a vector relative to the positive x-axis is given by:

$\theta = \tan^{-1}\left(\frac{v_y}{v_x}\right)$

Applications of Vectors

• Vectors are used to model real-world phenomena such as force, velocity, and displacement.

• They are essential in physics, engineering, and navigation.

Summary Table: Vector Operations

Operation	Graphical Representation	Algebraic Formula
Addition	Tip-to-tail method	$\vec{u} + \vec{v} = \langle u_x + v_x, u_y + v_y \rangle$
Subtraction	Reverse direction of subtracted vector, then add	$\vec{u} - \vec{v} = \langle u_x - v_x, u_y - v_y \rangle$
Magnitude	Length of arrow	$|\vec{v}| = \sqrt{v_x^2 + v_y^2}$
Direction	Angle from x-axis	$\theta = \tan^{-1}\left(\frac{v_y}{v_x}\right)$

Additional info: Vectors are foundational in trigonometry, especially in applications involving forces, navigation, and physics problems.