Learning Outcomes

Upon completing this lesson, you will be able to:

• Perform vector addition and scalar multiplication.
• Represent a 2D vector using Cartesian component notation.
• Resolve a vector into its rectangular components.
• Calculate the magnitude and direction of a vector.
• Define a unit vector and use it to describe a vector's direction.

Topics Covered

• Vector Operations
• Cartesian Vector Notation
• Magnitude, Direction, and Components
• Unit Vectors

I. Vector Operations

Since vectors have both magnitude and direction, the rules for their addition and multiplication are different from those for scalars.

A. Scalar Multiplication

Multiplying a vector $\vec{A}$ by a positive scalar $c$ results in a new vector $c\vec{A}$ that has the same direction as $\vec{A}$ but a magnitude that is $c$ times larger.

If $c$ is negative, the resulting vector $-c\vec{A}$ has a magnitude that is $|c|$ times larger, but its direction is reversed by $180^\circ$.

B. Vector Addition

The sum of two vectors $\vec{A}$ and $\vec{B}$ is a resultant vector $\vec{R} = \vec{A} + \vec{B}$. Graphically, this can be found using two primary methods:

1. Parallelogram Law: Place the tails of the two vectors at the same point. The resultant vector $\vec{R}$ is the diagonal of the parallelogram formed by the two vectors.

1. Triangle Rule (Tip-to-Tail): Place the tail of vector $\vec{B}$ at the tip of vector $\vec{A}$. The resultant vector $\vec{R}$ is drawn from the tail of $\vec{A}$ to the tip of $\vec{B}$.

Vector subtraction is defined as adding the negative:

$\vec{A} - \vec{B} = \vec{A} + (-\vec{B})$

II. Cartesian Vector Notation

While graphical methods are useful for visualization, analytical methods are required for accurate calculations. We use a rectangular coordinate system (typically $x$ and $y$) to represent vectors in terms of their components.

A. Unit Vectors ($\mathbf{i}$) and ($\mathbf{j}$)

We define two special vectors called unit vectors:

• $\mathbf{i}$ is a dimensionless vector of magnitude $1$ that points in the positive $x$-direction.
• $\mathbf{j}$ is a dimensionless vector of magnitude $1$ that points in the positive $y$-direction.

B. Component Form

Any vector $\vec{F}$ can be expressed as the sum of its two rectangular components, $F_x$ and $F_y$. Using the unit vectors $\mathbf{i}$ and $\mathbf{j}$, we can write the vector in Cartesian vector form:

$$ \vec{F} = F_x \mathbf{i} + F_y \mathbf{j} $$

• $F_x$ is the scalar component (magnitude) of the vector in the $x$-direction.
• $F_y$ is the scalar component (magnitude) of the vector in the $y$-direction.

This notation makes vector addition simple.

If $\vec{A} = A_x \mathbf{i} + A_y \mathbf{j}$ and $\vec{B} = B_x \mathbf{i} + B_y \mathbf{j}$, then:

$$ \vec{R} = \vec{A} + \vec{B} = (A_x + B_x)\mathbf{i} + (A_y + B_y)\mathbf{j} $$

III. Magnitude, Direction, and Components