🎯 Vectors (Light)
Basic vector operations in 2D for AMC 12. Focus on dot product, projections, and geometric applications.
🎯 Key Ideas
Vector: Directed line segment with magnitude and direction, represented as $\vec{v} = \langle a, b \rangle$ or $\vec{v} = a\mathbf{i} + b\mathbf{j}$.
Dot Product: $\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta = a_1b_1 + a_2b_2$.
Applications: Angle between vectors, projections, perpendicularity.
📐 Vector Basics
Notation
- Component form: $\vec{v} = \langle v_1, v_2 \rangle$
- Unit vectors: $\mathbf{i} = \langle 1, 0 \rangle$, $\mathbf{j} = \langle 0, 1 \rangle$
- Magnitude: $|\vec{v}| = \sqrt{v_1^2 + v_2^2}$
Vector Operations
- Addition: $\vec{u} + \vec{v} = \langle u_1 + v_1, u_2 + v_2 \rangle$
- Subtraction: $\vec{u} - \vec{v} = \langle u_1 - v_1, u_2 - v_2 \rangle$
- Scalar multiplication: $k\vec{v} = \langle kv_1, kv_2 \rangle$
Example: Find magnitude of $\vec{v} = \langle 3, -4 \rangle$
Solution:
- $|\vec{v}| = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5$
🔢 Dot Product
Definition
For vectors $\vec{a} = \langle a_1, a_2 \rangle$ and $\vec{b} = \langle b_1, b_2 \rangle$: $$\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2$$
Geometric Interpretation
$$\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta$$
where $\theta$ is the angle between the vectors.
Properties
- Commutative: $\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a}$
- Distributive: $\vec{a} \cdot (\vec{b} + \vec{c}) = \vec{a} \cdot \vec{b} + \vec{a} \cdot \vec{c}$
- Scalar multiplication: $(k\vec{a}) \cdot \vec{b} = k(\vec{a} \cdot \vec{b})$
Example: Find dot product of $\vec{a} = \langle 2, 3 \rangle$ and $\vec{b} = \langle -1, 4 \rangle$
Solution:
- $\vec{a} \cdot \vec{b} = 2(-1) + 3(4) = -2 + 12 = 10$
📐 Angle Between Vectors
Formula
$$\cos\theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|}$$
Special Cases
- Perpendicular: $\vec{a} \cdot \vec{b} = 0$ (angle = 90°)
- Parallel: $\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|$ (angle = 0°)
- Opposite: $\vec{a} \cdot \vec{b} = -|\vec{a}||\vec{b}|$ (angle = 180°)
Example: Find angle between $\vec{u} = \langle 1, 1 \rangle$ and $\vec{v} = \langle 1, 0 \rangle$
Solution:
- Dot product: $\vec{u} \cdot \vec{v} = 1(1) + 1(0) = 1$
- Magnitudes: $|\vec{u}| = \sqrt{1^2 + 1^2} = \sqrt{2}$, $|\vec{v}| = \sqrt{1^2 + 0^2} = 1$
- Angle: $\cos\theta = \frac{1}{\sqrt{2} \cdot 1} = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$
- Answer: $\theta = 45°$
📏 Projections
Scalar Projection
Projection of $\vec{a}$ onto $\vec{b}$: $$\text{proj}_{\vec{b}} \vec{a} = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|}$$
Vector Projection
$$\text{Proj}_{\vec{b}} \vec{a} = \left(\frac{\vec{a} \cdot \vec{b}}{|\vec{b}|^2}\right)\vec{b}$$
Example: Find projection of $\vec{a} = \langle 3, 4 \rangle$ onto $\vec{b} = \langle 1, 0 \rangle$
Solution:
- Dot product: $\vec{a} \cdot \vec{b} = 3(1) + 4(0) = 3$
- Magnitude of $\vec{b}$: $|\vec{b}| = \sqrt{1^2 + 0^2} = 1$
- Scalar projection: $\text{proj}_{\vec{b}} \vec{a} = \frac{3}{1} = 3$
- Vector projection: $\text{Proj}_{\vec{b}} \vec{a} = \frac{3}{1^2} \langle 1, 0 \rangle = \langle 3, 0 \rangle$
🔄 Unit Vectors
Definition
Vector with magnitude 1: $|\hat{u}| = 1$
Finding Unit Vector
$$\hat{u} = \frac{\vec{u}}{|\vec{u}|}$$
Example: Find unit vector in direction of $\vec{v} = \langle 3, -4 \rangle$
Solution:
- Magnitude: $|\vec{v}| = \sqrt{3^2 + (-4)^2} = 5$
- Unit vector: $\hat{v} = \frac{\langle 3, -4 \rangle}{5} = \langle \frac{3}{5}, -\frac{4}{5} \rangle$
🎯 AMC-Style Worked Example
Problem: Find the area of the parallelogram formed by vectors $\vec{u} = \langle 2, 3 \rangle$ and $\vec{v} = \langle 1, 4 \rangle$.
Solution:
- Cross product magnitude: For 2D vectors $\vec{u} = \langle u_1, u_2 \rangle$ and $\vec{v} = \langle v_1, v_2 \rangle$, the cross product magnitude is $|u_1v_2 - u_2v_1|$
- Calculate: $|2 \cdot 4 - 3 \cdot 1| = |8 - 3| = 5$
- Area: The area of the parallelogram is equal to the magnitude of the cross product
Answer: 5
🔍 Common Traps & Fixes
Trap: Confusing dot product and cross product
Fix: Dot product gives scalar: $\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2$; cross product gives vector (in 3D) or scalar magnitude (in 2D).
Trap: Wrong angle formula
Fix: $\cos\theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|}$, not $\sin\theta$.
Trap: Forgetting absolute value in projection
Fix: Scalar projection can be negative (indicates opposite direction), but magnitude is always positive.
Trap: Wrong unit vector calculation
Fix: Divide by magnitude, not by sum of components.
📋 Quick Reference
Vector Operations
- Magnitude: $|\vec{v}| = \sqrt{v_1^2 + v_2^2}$
- Addition: $\vec{u} + \vec{v} = \langle u_1 + v_1, u_2 + v_2 \rangle$
- Scalar multiplication: $k\vec{v} = \langle kv_1, kv_2 \rangle$
Dot Product
- Component form: $\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2$
- Geometric: $\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta$
- Perpendicular: $\vec{a} \cdot \vec{b} = 0$
Projections
- Scalar: $\text{proj}_{\vec{b}} \vec{a} = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|}$
- Vector: $\text{Proj}_{\vec{b}} \vec{a} = \left(\frac{\vec{a} \cdot \vec{b}}{|\vec{b}|^2}\right)\vec{b}$
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