Vectors: Definitions, Operations, and Properties in Calculus

Vectors in Calculus

Definition and Basic Properties

Vectors are mathematical objects that possess both magnitude and direction. They are fundamental in describing quantities such as displacement, velocity, and force in calculus and physics.

• Definition: A vector is an ordered pair or tuple, often written as $\vec{a} = \langle x, y \rangle$ in two dimensions.

• Remark: Vectors can describe the relative position between points in space.

Vectors in the Coordinate System

Vectors are often represented in a coordinate system, where their components correspond to their projections along the axes.

• Position Vector: The vector $\vec{OP}$ from the origin $O$ to a point $P(x, y)$ is $\langle x, y \rangle$.

• Example: For points $P = (2, 2)$ and $Q = (3, -1)$, the vector $\vec{PQ}$ is $\langle 1, -3 \rangle$.

• Vector Addition: $\vec{OP} + \vec{PQ} = \vec{OQ}$

Vector Operations

Vector Addition

Vectors can be added together to produce a new vector. This operation is performed componentwise.

• Algebraic Rule: $\langle x_1, y_1 \rangle + \langle x_2, y_2 \rangle = \langle x_1 + x_2, y_1 + y_2 \rangle$

• Example: If $\vec{a} = \langle 2, 2 \rangle$ and $\vec{b} = \langle 1, -3 \rangle$, then $\vec{a} + \vec{b} = \langle 3, -1 \rangle$.

• Commutativity: Vector addition is commutative: $\vec{a} + \vec{b} = \vec{b} + \vec{a}$.

Vector Subtraction

Subtracting vectors is also performed componentwise.

• Algebraic Rule: $\vec{c} = \vec{a} - \vec{b}$

• Example: If $\vec{a} = \langle 2, 2 \rangle$ and $\vec{b} = \langle 1, -3 \rangle$, then $\vec{a} - \vec{b} = \langle 1, 5 \rangle$.

Scalar Multiplication

Multiplying a vector by a scalar changes its magnitude but not its direction (unless the scalar is negative).

• Algebraic Rule: $m \langle x, y \rangle = \langle m x, m y \rangle$

• Examples:

  • $3 \vec{a} = \langle 6, 6 \rangle$ if $\vec{a} = \langle 2, 2 \rangle$

  • $\frac{1}{2} \vec{a} = \langle 1, 1 \rangle$

  • $(-1) \vec{a} = \langle -2, -2 \rangle$

  • $0 \vec{a} = \langle 0, 0 \rangle$ (the zero vector)

Special Notations

• Negative Vector: $-\vec{a}$

• Zero Vector: $\vec{0}$

Magnitude and Length of Vectors

Pythagorean Theorem for Vectors

The magnitude or length of a vector $\vec{v} = \langle x, y \rangle$ is given by:

• $|\vec{v}| = \sqrt{x^2 + y^2}$

• Example: For $\vec{v} = \langle 3, 4 \rangle$, $|\vec{v}| = \sqrt{3^2 + 4^2} = 5$.

Unit Vectors

A unit vector is a vector with length 1. Unit vectors are often used to indicate direction.

• Special Unit Vectors:

  • $\hat{i} = \langle 1, 0 \rangle$ (x-direction)

  • $\hat{j} = \langle 0, 1 \rangle$ (y-direction)

Distance Between Two Points

The distance between two points $P = (x_1, y_1)$ and $Q = (x_2, y_2)$ is the length of the vector $\vec{PQ}$:

• $|\vec{PQ}| = |\vec{OQ} - \vec{OP}| = |\langle x_2 - x_1, y_2 - y_1 \rangle|$

• $|\vec{PQ}| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Summary Table: Vector Operations

Operation	Algebraic Rule	Example
Addition	$\langle x_1, y_1 \rangle + \langle x_2, y_2 \rangle = \langle x_1 + x_2, y_1 + y_2 \rangle$	$\langle 2, 2 \rangle + \langle 1, -3 \rangle = \langle 3, -1 \rangle$
Subtraction	$\langle x_1, y_1 \rangle - \langle x_2, y_2 \rangle = \langle x_1 - x_2, y_1 - y_2 \rangle$	$\langle 2, 2 \rangle - \langle 1, -3 \rangle = \langle 1, 5 \rangle$
Scalar Multiplication	$m \langle x, y \rangle = \langle m x, m y \rangle$	$3 \langle 2, 1 \rangle = \langle 6, 3 \rangle$
Magnitude	$|\langle x, y \rangle| = \sqrt{x^2 + y^2}$	$|\langle 3, 4 \rangle| = 5$

Additional info:

• Vectors are foundational for multivariable calculus, physics, and engineering.

• Understanding vector operations is essential for topics such as vector calculus, dot product, and cross product.