Quick Reference

Coordinate Systems and Components of a Vector

• Unit Vectors

• Vectors in Two Dimensions

where

• is a vector in two dimensions
• is the vector component in the -direction
• is the vector component in the -direction

where:

• is the scalar component of
• is the scalar component of

where:

• are the coordinates of the origin point of a vector.

• are the coordinates of the end point of a vector

• Magnitude of a 2D Vector

where

• is the magnitude of vector .

• Vector Trigonometry

where:

• is the angle between the vector components and .

• Polar Coordinates

where:

• is a point in the Cartesian plane

• is the radial coordinate

• is the angle that the vector makes with the unit radial vector.

• Vectors in Three Dimensions

where:

• is a 3D vector
• is the vector component in the -direction

where:

• is the scalar component of

where:

• is the coordinates of the origin point of a vector.
• is the coordinates of the end point of a vector.

where:

• is the magnitude of the 3D vector .

Algebra of Vectors

• Scalar Multiplication

• Antiparallel Vector
  • Two vectors that points in opposite directions/points opposite to one another.

• Null Vector
  • A vector with all components at zero.

• Equal Vectors (a)
  • Two vectors and are equal if:

• Equal Vectors (b)
  • Any two vectors and are equal if their difference is the null vector.

• Addition of Two Vectors
  • Adding two vectors is the same as adding their corresponding components.

• Addition of Multiple Vectors
  • We can expand this idea to vectors.

• Unit Vector of Direction
  • The unit vector of direction is a vector with magnitude but also points to the direction of any vector

Product of Vectors

• Scalar/Dot Product of Vectors

where:

• is the angle between the vectors and .

• Properties of the Dot Product

  • For certain values of , the dot product of two vectors may have some special values.

• Dot Product of Unit Vectors
  • Since unit vectors are orthogonal, then their dot products are .

• Dot Product of a Unit Vector to Itself
  • Since the magnitude of a unit vector is 1, the dot product of a unit vector to itself is always .

• Dot Product of a Unit Vector to a Vector
  • The dot product of a vector to a unit vector is the magnitude of the component vector in that direction.

• Dot Product of Two Vectors in Component Form

• Angle of Two Vectors using Dot Product