The Main Idea 

Some quantities need just a number—like temperature or distance. These are scalars. But other quantities need both a number and a direction to make sense. For example, saying “the wind is blowing at 15 mph” is incomplete without knowing which direction. That’s where vectors come in.

A vector has both magnitude (size or strength) and direction. We write them in bold like [latex]\mathbf{v}[/latex], or with an arrow when handwriting: [latex]\overrightarrow{v}[/latex].

Think of a vector as an arrow on a graph. It starts at the initial point and ends at the terminal point. The arrow’s length shows magnitude ([latex]||\mathbf{v}||[/latex]), and the arrow points in the vector’s direction.

Key Concepts

• Zero vector ([latex]\mathbf{0}[/latex]): Initial and terminal points are the same—zero magnitude, no specific direction.
• Equivalent vectors: Vectors with the same magnitude and direction are equal ([latex]\mathbf{v} = \mathbf{w}[/latex]), even if they start at different points. Think of it like two people walking 5 miles north—same journey, different starting locations.

Quick Check for Equivalence

Ask: Same length? Same direction? If yes to both, they’re equivalent—starting location doesn’t matter.

The Main Idea 

Working with vectors geometrically (drawing arrows and parallelograms) works, but using coordinates makes calculations much easier and more precise.

Any vector can be written as [latex]\mathbf{v} = \langle x, y \rangle[/latex], where [latex]x[/latex] and [latex]y[/latex] are its components. This notation describes a vector starting at the origin [latex](0,0)[/latex] and ending at [latex](x,y)[/latex].

Notation: Use angle brackets [latex]\langle x, y \rangle[/latex] for vectors, not parentheses [latex](x,y)[/latex]. Parentheses describe points; angle brackets describe vectors.

Converting to Component Form

For a vector with initial point [latex](x_1, y_1)[/latex] and terminal point [latex](x_2, y_2)[/latex]:

[latex]\mathbf{v} = \langle x_2 - x_1, y_2 - y_1 \rangle[/latex]

Think “terminal minus initial” for each coordinate. This gives you the horizontal and vertical change from start to finish.

Finding Magnitude

The magnitude (length) of [latex]\mathbf{v} = \langle x, y \rangle[/latex] uses the distance formula:

[latex]||\mathbf{v}|| = \sqrt{x^2 + y^2}[/latex]

This comes from the Pythagorean theorem—the components form the legs of a right triangle, and the magnitude is the hypotenuse.

From Magnitude and Direction to Components

If you know a vector’s magnitude [latex]||\mathbf{v}||[/latex] and direction angle [latex]\theta[/latex] (measured from the positive [latex]x[/latex]-axis), use trigonometry:

[latex]\mathbf{v} = \langle ||\mathbf{v}|| \cos \theta, ||\mathbf{v}|| \sin \theta \rangle[/latex]

The cosine gives the horizontal component; the sine gives the vertical component.

The Main Idea 

A unit vector has magnitude exactly [latex]1[/latex]. Unit vectors are useful because they capture direction without being tied to any specific size—think of them as “direction indicators.”

Creating a Unit Vector (Normalization)

To find a unit vector [latex]\mathbf{u}[/latex] pointing in the same direction as any nonzero vector [latex]\mathbf{v}[/latex], divide [latex]\mathbf{v}[/latex] by its magnitude:

[latex]\mathbf{u} = \frac{1}{||\mathbf{v}||}\mathbf{v}[/latex]

This process is called normalization. You’re essentially scaling the vector down (or up) to length [latex]1[/latex] while preserving its direction.

Why This Works: Dividing by [latex]||\mathbf{v}||[/latex] is just scalar multiplication by [latex]\frac{1}{||\mathbf{v}||[/latex], which changes magnitude but not direction.

Building Vectors from Unit Vectors

Once you have a unit vector [latex]\mathbf{u}[/latex] in the desired direction, you can create any vector in that direction by scalar multiplication. For example, [latex]7\mathbf{u}[/latex] has magnitude [latex]7[/latex] and points the same way as [latex]\mathbf{u}[/latex].

Standard Unit Vectors [latex]\mathbf{i}[/latex] and [latex]\mathbf{j}[/latex]

Two special unit vectors point along the coordinate axes:

• [latex]\mathbf{i} = \langle 1, 0 \rangle[/latex] (horizontal, along positive [latex]x[/latex]-axis)
• [latex]\mathbf{j} = \langle 0, 1 \rangle[/latex] (vertical, along positive [latex]y[/latex]-axis)

Linear Combination Form

Any vector [latex]\mathbf{v} = \langle x, y \rangle[/latex] can be written using [latex]\mathbf{i}[/latex] and [latex]\mathbf{j}[/latex]:

[latex]\mathbf{v} = x\mathbf{i} + y\mathbf{j}[/latex]

This shows [latex]\mathbf{v}[/latex] as the sum of a horizontal component ([latex]x\mathbf{i}[/latex]) and a vertical component ([latex]y\mathbf{j}[/latex]).

Unit Vectors from Angles

If a unit vector makes angle [latex]\theta[/latex] with the positive [latex]x[/latex]-axis, its components come directly from the unit circle:

[latex]\mathbf{u} = \langle \cos \theta, \sin \theta \rangle = (\cos \theta)\mathbf{i} + (\sin \theta)\mathbf{j}[/latex]