Algebraic Vectors in 2- and 3-Space

Algebraic Vectors in 2- and 3-Space

• Vector Representation: Vectors can be represented algebraically. A vector AB in two dimensions is expressed as (x₂ - x₁, y₂ - y₁), where (x₁, y₁) and (x₂, y₂) are coordinate points.

• Unit Vectors: A unit vector is parallel to a coordinate axis (x, y, or z) and has a length of 1.

  • In two dimensions: i = (1, 0) and j = (0, 1)
  • In three dimensions: i = (1, 0, 0), j = (0, 1, 0), and k = (0, 0, 1)

• Vector Magnitude (Length): The length of a vector is calculated using the Pythagorean theorem.

Vector Components and Direction

• Component Form: A vector's component form uses its magnitude and direction angle to find its horizontal and vertical components.

• Direction Angle: Direction angles are used to find the angle of a vector from the horizontal/positive x-axis.

• Magnitude and Direction: A vector's magnitude (or length) and its direction angle fully describe the vector.

Unit Vectors

• Definitions: Unit vectors have length 1 and share the same direction as another vector.

  • Finding unit vectors is about scaling a vector down to a unit length without changing its direction.

• Formula: A unit vector in the same direction as vector V is (V/|V|)

Problems (Examples from the notes)

• Force Components: Finding the horizontal and vertical components of a force vector with given magnitude and direction angle.

• Vector Addition: Calculating the resultant force (vector sum) when two or more forces act simultaneously.

• Equivalent Vectors: Determining if two vectors (e.g., AB and CD) are the same, meaning they have the same magnitude and direction.

• Determining unit vectors: Finding a unit vector with the same direction as a given vector V.

• Calculating directed distance: Finding the ordered pair of a vector given its magnitude and direction.

• Direction angle of a given vector: Finding the direction angle of a given vector, commonly expressed in degrees from the x-axis.