VECTORS Dr. Shildneck

Vectors There are two types of quantities in the world. Scalar – a quantity that is specified by a single value with an appropriate unit and has no direction. (Examples: temperature, height, length of string) Vector – a quantity that has both magnitude (size) and direction. (Examples: driving west at 50 mph, pulling a cart up a hill, weight)

Vectors Vectors have non-negative magnitude (size) and a specific direction. To represent them, we use directed line segments. The segments have an initial point and a terminal point. At the terminal point, we represent the direction of the vector with an “arrow head.” P Q Example: The vector from P to Q.

Notations Notations for Vectors using points: (use a half arrow over the points in order) using vector name (typed): v (bold lowercase letter) using vector name (handwritten): (lowercase letter with half arrow) Notation for Magnitude

Equal Vectors Vectors are equal if they have the same magnitude AND direction. Location does not matter when determining if vectors are equal. To show that two vectors are equal, show that their magnitude is the same and that they travel in the same direction.

Scalar Multiplication – Resizing Vectors Any vector can be resized by multiplying it by a real number (scalar). Multiplying by positive scalar changes magnitude only. Multiplying by a negative scalar changes the magnitude and its direction.

Adding Vectors – Geometrically “Parallelogram Method” Given two vectors, to add them geometrically, you can use a parallelogram. First, join the vectors initial points (tails). Second, create two equal vectors whose tails meet the heads of the first set and whose heads join. This forms a parallelogram. Finally, the resultant vector of this addition is the diagonal from the joined tails to the joined heads.

Adding Vectors – Geometrically “Parallelogram Method” Given two vectors, to add them geometrically, you can use a parallelogram. First, join the vectors initial points (tails). Second, create two equal vectors whose tails meet the heads of the first set and whose heads join. This forms a parallelogram. Finally, the resultant vector of this addition is the diagonal from the joined tails to the joined heads.

Adding Vectors – Geometrically “Parallelogram Method” u v u + v

Component Form The component form of a vector is written as the “end point” when the vector is in standard position (initial point at the origin). The component form of the vector from P(a, b) to Q(c, d) can be found by subtracting the components of each point. This is also called the position vector.

Linear Combinations The linear combination form of a vector uses scalars of the standard unit vectors i = and j = to write the position vector. For example, the vector can be written in the form 3i + 5j Because 3i + 5j = = + =

Resizing Written Vectors As you could see in the example for Linear Combinations, vectors can be resized by multiplying by scalars. Given u =, 4u = 4 = =

Adding Vectors in Written Form Adding vectors in written forms is fairly simple. Basically you just have to follow the order of operations. In component form: 1. Multiply through by any scalars. 2. Add horizontal components, Add vertical components In Linear combinations: 1. Combine like terms.

Unit Vectors A unit vector is a vector of magnitude 1 (in any direction). To find a unit vector in a specific direction, you must “divide” a given vector using scalar multiplication so that the new vector’s magnitude is 1. Think about what you would have to multiply by to make this happen.

Assignment 1. Vector Concepts #1 (WS) 2. Vector Concepts #2 (WS) 3. Intro to Vectors WS