Vector & Algebra Cheatsheet

Cheat Codes & Shortcuts

• Vector: A quantity with both magnitude and direction, e.g. \( \mathbf{v} = \langle v_1, v_2, v_3 \rangle \).
• Vector Addition: \( \mathbf{a} + \mathbf{b} = \langle a_1 + b_1, a_2 + b_2, a_3 + b_3 \rangle \).
• Dot Product: \( \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3 \); gives scalar.
• Cross Product: \( \mathbf{a} \times \mathbf{b} = \langle a_2b_3 - a_3b_2, a_3b_1 - a_1b_3, a_1b_2 - a_2b_1 \rangle \); vector perpendicular to both.
• Magnitude: \( |\mathbf{v}| = \sqrt{v_1^2 + v_2^2 + v_3^2} \).
• Algebraic Expressions: Use distributive, associative, commutative laws to simplify.
• Factoring: \( ax^2 + bx + c = 0 \Rightarrow (mx + n)(px + q) = 0 \).
• Quadratic Formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
• Polynomials: Degree indicates highest power; factor or use synthetic division.
• Linear Equations: \( ax + b = 0 \Rightarrow x = -\frac{b}{a} \).

Quick Reference Table

Topic	Formula/Concept	Notes
Vector Addition	\( \mathbf{a} + \mathbf{b} = \langle a_1 + b_1, a_2 + b_2, a_3 + b_3 \rangle \)	Component-wise addition
Dot Product	\( \mathbf{a} \cdot \mathbf{b} = \sum a_ib_i \)	Scalar product; \( |\mathbf{a}| |\mathbf{b}| \cos \theta \)
Cross Product	\( \mathbf{a} \times \mathbf{b} \) as determinant	Vector perpendicular to \( \mathbf{a} \) and \( \mathbf{b} \)
Quadratic Formula	\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)	Solves \( ax^2 + bx + c = 0 \)
Polynomial Factor	Factor by grouping, synthetic division	Important for solving polynomial equations
Linear Equation	\( ax + b = 0 \)	Isolate \( x \)

Advice

Vector & Algebra Quick Tips

• Vector Magnitude: Use \( \sqrt{x^2 + y^2 + z^2} \) for length.
• Dot Product Zero: Means vectors are perpendicular (orthogonal).
• Cross Product Direction: Use right-hand rule for resultant vector direction.
• Quadratic Roots: Use discriminant \( b^2 - 4ac \) to determine solution type.
• Factor First: Always try to factor polynomials before using formulas.

Vector & Algebra Speed Quiz

Test your speed with 5 vector and algebra questions! You have 30 seconds per question.