Vectors in Component Form: Concepts and Examples

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 component form, which makes calculations involving addition, subtraction, and magnitude straightforward.

• Component Form: A vector in the plane can be written as v = <v1, v2>, where v1 and v2 are the horizontal and vertical components, respectively.

• Magnitude: The magnitude (or length) of a vector v = <a, b> is given by: $|\vec{v}| = \sqrt{a^2 + b^2}$

Vector Addition and Subtraction

Vectors can be added or subtracted by combining their corresponding components.

• Addition: If u = <u1, u2> and v = <v1, v2>, then: $\vec{u} + \vec{v} = \langle u_1 + v_1,\ u_2 + v_2 \rangle$

• Subtraction: If u = <u1, u2> and v = <v1, v2>, then: $\vec{u} - \vec{v} = \langle u_1 - v_1,\ u_2 - v_2 \rangle$

Examples

• Example 1: Given u = <3, 5> and v = <-1, 2>, find u + v and u - v.

  • $\vec{u} + \vec{v} = \langle 3 + (-1),\ 5 + 2 \rangle = \langle 2, 7 \rangle$

  • $\vec{u} - \vec{v} = \langle 3 - (-1),\ 5 - 2 \rangle = \langle 4, 3 \rangle$

• Example 2: Given u = <2, -4> and v = <5, 1>, find u + v and u - v.

  • $\vec{u} + \vec{v} = \langle 2 + 5,\ -4 + 1 \rangle = \langle 7, -3 \rangle$

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

Finding the Magnitude of a Vector

The magnitude of a vector v = <a, b> is calculated using the Pythagorean theorem:

$|\vec{v}| = \sqrt{a^2 + b^2}$

• Example: For v = <5, 7>:

  • $|\vec{v}| = \sqrt{5^2 + 7^2} = \sqrt{25 + 49} = \sqrt{74}$

Summary Table: Vector Operations

Operation	Formula	Example
Addition	$\langle a, b \rangle + \langle c, d \rangle = \langle a + c,\ b + d \rangle$	$\langle 2, -4 \rangle + \langle 5, 1 \rangle = \langle 7, -3 \rangle$
Subtraction	$\langle a, b \rangle - \langle c, d \rangle = \langle a - c,\ b - d \rangle$	$\langle 2, -4 \rangle - \langle 5, 1 \rangle = \langle -3, -5 \rangle$
Magnitude	$|\langle a, b \rangle| = \sqrt{a^2 + b^2}$	$|\langle 5, 7 \rangle| = \sqrt{74}$

Additional info: Vectors are foundational in trigonometry and physics, as they allow for the representation of quantities that have both magnitude and direction, such as force, velocity, and displacement.