Vectors: Representation and Operations in Component Form

Vectors in Component Form

Introduction to Vectors

Vectors are mathematical objects that have both magnitude and direction. In trigonometry and precalculus, vectors are often represented in the coordinate plane using component form, which allows for straightforward calculations and visualizations.

• Component Form: A vector from point A $(x_1, y_1)$ to point B $(x_2, y_2)$ is written as $\langle x_2 - x_1, y_2 - y_1 \rangle$.

• Notation: Vectors are commonly denoted by boldface letters (e.g., v) or with an arrow above (e.g., $\vec{v}$).

Graphical Representation of Vectors

Vectors can be visualized as arrows in the coordinate plane. The tail of the arrow is at the initial point, and the head is at the terminal point.

• Example: The vector from $(1, 2)$ to $(4, 5)$ is $\langle 4-1, 5-2 \rangle = \langle 3, 3 \rangle$.

• Standard Position: A vector is in standard position if its initial point is at the origin $(0, 0)$.

Operations with Vectors in Component Form

Vectors in component form can be added, subtracted, and multiplied by scalars using simple arithmetic operations.

• Addition: $\langle a, b \rangle + \langle c, d \rangle = \langle a + c, b + d \rangle$

• Subtraction: $\langle a, b \rangle - \langle c, d \rangle = \langle a - c, b - d \rangle$

• Scalar Multiplication: $k \langle a, b \rangle = \langle ka, kb \rangle$

Example Problem

Given the vector $\vec{v}$ from $A = (2, 1)$ to $B = (5, 4)$:

• Component Form: $\vec{v} = \langle 5-2, 4-1 \rangle = \langle 3, 3 \rangle$

• Magnitude: $|\vec{v}| = \sqrt{3^2 + 3^2} = \sqrt{18} = 3\sqrt{2}$

Summary Table: Vector Operations

Operation	Formula	Example
Addition	$\langle a, b \rangle + \langle c, d \rangle = \langle a + c, b + d \rangle$	$\langle 2, 3 \rangle + \langle 1, 4 \rangle = \langle 3, 7 \rangle$
Subtraction	$\langle a, b \rangle - \langle c, d \rangle = \langle a - c, b - d \rangle$	$\langle 5, 6 \rangle - \langle 2, 1 \rangle = \langle 3, 5 \rangle$
Scalar Multiplication	$k \langle a, b \rangle = \langle ka, kb \rangle$	$2 \langle 3, 4 \rangle = \langle 6, 8 \rangle$

Applications

• Vectors are used in physics to represent quantities such as force and velocity.

• In trigonometry, vectors help solve problems involving direction and magnitude in the coordinate plane.

Additional info: Vectors and their operations are foundational for later topics in trigonometry, including applications to physics and engineering problems.