Vector Algebra Cheat Sheet

This cheat sheet covers fundamental concepts of vectors, including their definition, types, operations (addition, subtraction, dot product, cross product), and applications in physics.

Core Principles

• Vectors have both magnitude and direction, unlike scalars which only have magnitude.
• Vector addition and subtraction follow specific rules (e.g., parallelogram law, triangle law).
• The dot product of two vectors results in a scalar quantity, while the cross product results in a vector quantity.
• Unit vectors (i, j, k) are mutually perpendicular vectors of unit magnitude, used to represent directions along axes.
• The magnitude of a vector A = Ax i + Ay j + Az k is given by |A| = sqrt(Ax^2 + Ay^2 + Az^2).
• The scalar triple product A . (B x C) represents the volume of a parallelepiped formed by the three vectors.

Formulas

• Vector Magnitude: |A| = sqrt(Ax^2 + Ay^2 + Az^2)
• Dot Product: A · B = |A||B|cosθ = AxBx + AyBy + AzBz
• Cross Product: |A × B| = |A||B|sinθ
• Scalar Triple Product: A · (B × C)
• Vector Triple Product: A × (B × C) = (A · C)B – (A · B)C

Quiz

• If |A + B| = |A - B|, what is the angle between vectors A and B?: 90°
• The dot product of two vectors is a:: Scalar
• Which of the following is a unit vector?: i + j