AP Precalculus: Two-Dimensional Vectors

Master vector operations, magnitude, direction, and component form

📐 Magnitude 🎯 Direction ➕ Operations 📏 Unit Vectors

📚 Understanding Vectors

A vector is a quantity with both magnitude (length) and direction. In two dimensions, vectors can be represented as arrows on a coordinate plane or written in component form. Vectors are essential in physics, engineering, and computer graphics for representing forces, velocities, and displacements.

1 Vector Notation & Basics

A vector in 2D can be written in component form as \(\vec{v} = \langle a, b \rangle\), where \(a\) is the horizontal component and \(b\) is the vertical component.

Component Form

\(\vec{v} = \langle a, b \rangle\)

Horizontal Component

\(a\) (x-direction)

Vertical Component

\(b\) (y-direction)

Standard Unit Vectors

\(\mathbf{i} = \langle 1, 0 \rangle\), \(\mathbf{j} = \langle 0, 1 \rangle\)

💡 Alternative Notation

\(\vec{v} = \langle a, b \rangle = a\mathbf{i} + b\mathbf{j}\). Both forms are equivalent!

2 Component Form from Two Points

Given two points, find the vector from the initial point to the terminal point by subtracting coordinates.

Vector from A to B \(\vec{AB} = \langle x_2 - x_1, y_2 - y_1 \rangle\)

where \(A(x_1, y_1)\) is the initial point and \(B(x_2, y_2)\) is the terminal point

📌 Example

Find: Vector from \(A(2, 3)\) to \(B(7, -1)\)

Calculate: \(\vec{AB} = \langle 7 - 2, -1 - 3 \rangle = \langle 5, -4 \rangle\)

3 Magnitude of a Vector

The magnitude (or length) of a vector is its distance from the origin, calculated using the Pythagorean theorem.

Magnitude Formula \(|\vec{v}| = |\langle a, b \rangle| = \sqrt{a^2 + b^2}\)

📌 Example

Find: \(|\langle 3, -4 \rangle|\)

Calculate: \(\sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5\)

4 Direction Angle

The direction angle \(\theta\) is the angle the vector makes with the positive x-axis, measured counterclockwise.

Direction Angle Formula \(\theta = \arctan\left(\frac{b}{a}\right)\) (adjust for quadrant)

⚠️ Quadrant Adjustment

The basic arctan formula only gives angles in Quadrants I and IV. Add \(180°\) (or \(\pi\)) for Quadrants II and III!

📌 Example

Find direction of: \(\vec{v} = \langle -3, 3 \rangle\)

Reference angle: \(\arctan\left|\frac{3}{-3}\right| = \arctan(1) = 45°\)

Quadrant: \(a < 0, b> 0\) → Quadrant II

Direction: \(\theta = 180° - 45° = 135°\)

5 Component Form from Magnitude & Direction

Given magnitude \(r\) and direction angle \(\theta\), find the component form of the vector.

Component Form \(\vec{v} = \langle r\cos\theta, r\sin\theta \rangle\)

📌 Example

Given: Magnitude = 6, Direction = 60°

Components: \(\vec{v} = \langle 6\cos 60°, 6\sin 60° \rangle = \langle 6 \cdot \frac{1}{2}, 6 \cdot \frac{\sqrt{3}}{2} \rangle = \langle 3, 3\sqrt{3} \rangle\)

6 Vector Addition & Subtraction

Add or subtract vectors by combining their corresponding components.

Addition

\(\langle a, b \rangle + \langle c, d \rangle = \langle a+c, b+d \rangle\)

Add corresponding components

Subtraction

\(\langle a, b \rangle - \langle c, d \rangle = \langle a-c, b-d \rangle\)

Subtract corresponding components

Geometric Methods

Triangle Method (Tip-to-Tail)

Draw the first vector
Place the tail of the second vector at the tip of the first
The resultant goes from the start of the first to the end of the second

Parallelogram Method

Draw both vectors from the same point
Complete the parallelogram
The diagonal from the common point is the resultant

📌 Example

Add: \(\langle 4, -2 \rangle + \langle -1, 5 \rangle\)

Sum: \(\langle 4 + (-1), -2 + 5 \rangle = \langle 3, 3 \rangle\)

7 Magnitude & Direction of Resultant

After adding vectors to get the resultant, find its magnitude and direction using the standard formulas.

Resultant Magnitude

\(|\vec{R}| = \sqrt{(a+c)^2 + (b+d)^2}\)

Resultant Direction

\(\theta_R = \arctan\left(\frac{b+d}{a+c}\right)\)

📌 Example

Given: \(\vec{R} = \langle 3, 3 \rangle\) (from previous example)

Magnitude: \(|\vec{R}| = \sqrt{3^2 + 3^2} = \sqrt{18} = 3\sqrt{2}\)

Direction: \(\theta = \arctan\left(\frac{3}{3}\right) = \arctan(1) = 45°\)

8 Scalar Multiplication

Scalar multiplication multiplies each component by a constant (scalar), changing the vector's length but not its direction (unless negative).

Scalar Multiplication \(k \cdot \langle a, b \rangle = \langle ka, kb \rangle\)

If \(k > 0\)

Same direction, magnitude multiplied by \(k\)

If \(k < 0\)

Opposite direction, magnitude = \(|k| \cdot |\vec{v}|\)

If \(|k| > 1\)

Vector is stretched (longer)

If \(0 < |k| < 1\)

Vector is compressed (shorter)

📌 Example

Calculate: \(-2 \cdot \langle 3, -4 \rangle = \langle -6, 8 \rangle\)

Effect: Doubled in length, reversed direction

9 Unit Vector

A unit vector has magnitude 1 and points in the same direction as the original vector.

Unit Vector Formula \(\hat{u} = \frac{\vec{v}}{|\vec{v}|} = \frac{1}{|\vec{v}|} \langle a, b \rangle = \left\langle \frac{a}{|\vec{v}|}, \frac{b}{|\vec{v}|} \right\rangle\)

📌 Example

Find unit vector in direction of: \(\vec{v} = \langle 3, 4 \rangle\)

Magnitude: \(|\vec{v}| = 5\)

Unit vector: \(\hat{u} = \frac{1}{5}\langle 3, 4 \rangle = \left\langle \frac{3}{5}, \frac{4}{5} \right\rangle\)

💡 Verify Your Answer

Check that \(|\hat{u}| = 1\): \(\sqrt{\left(\frac{3}{5}\right)^2 + \left(\frac{4}{5}\right)^2} = \sqrt{\frac{9+16}{25}} = 1\) ✓

10 Linear Combinations

A linear combination of vectors combines scalar multiples of multiple vectors.

Linear Combination \(a\vec{v} + b\vec{w} = a\langle v_1, v_2 \rangle + b\langle w_1, w_2 \rangle = \langle av_1 + bw_1, av_2 + bw_2 \rangle\)

📌 Example

Calculate: \(2\langle 1, 3 \rangle + 3\langle -2, 1 \rangle\)

= \(\langle 2, 6 \rangle + \langle -6, 3 \rangle\)

= \(\langle 2 + (-6), 6 + 3 \rangle = \langle -4, 9 \rangle\)

📋 Quick Reference

Magnitude

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

Direction

\(\theta = \arctan\left(\frac{b}{a}\right)\)

Components from Polar

\(\langle r\cos\theta, r\sin\theta \rangle\)

Vector Addition

\(\langle a+c, b+d \rangle\)

Scalar Multiply

\(k\langle a, b \rangle = \langle ka, kb \rangle\)

Unit Vector

\(\hat{u} = \frac{\vec{v}}{|\vec{v}|}\)

📚 Understanding Vectors

1 Vector Notation & Basics

2 Component Form from Two Points

3 Magnitude of a Vector

4 Direction Angle

5 Component Form from Magnitude & Direction

6 Vector Addition & Subtraction

Geometric Methods

7 Magnitude & Direction of Resultant

8 Scalar Multiplication

If \(k > 0\)

If \(k < 0\)

If \(|k| > 1\)

If \(0 < |k| < 1\)

9 Unit Vector

10 Linear Combinations

📋 Quick Reference

Magnitude

Direction

Components from Polar

Vector Addition

Scalar Multiply

Unit Vector

Need Help with Vectors?