Vectors

Vectors and Scalars

Some text here about the topic of discussion

Vectors and Scalars

• Scalars are quantities that only have a magnitude.

  • Magnitude is the absolute quantity of an object.

  • Examples: Distance, Speed, Time, Energy

• Vectors are quantities that have magnitude and direction.

  • Distance is the direction that the subject will travel.

  • Examples: Displacement, Acceleration, Weight, Force

• A vector can be multiplied by a number (scalar)

  • For example: In the diagram, the vector (v) can be multiplied by ½, 3, or -2.

    • Notice how (v) changes direction when multiplied by -2​

Multipication of a vector by a scalar quantity

Addition of Vectors

• Vector addition and subtraction can be done by the parallelogram method or the head to tail method. Vectors that form a closed polygon (cycle) add up to zero

• For the parallelogram method:

1. Draw them at some common point (O)

2. Complete the sides of the parallelogram

3. Then draw the diagonal, this is the result of the addition of the two vectors

Subtraction of Vectors

• Similar to the parallelogram method

  1. Draw the vectors at a common point

  2. The vector from the tip of -w to v is the result of the subtraction

• It's like adding v and -w

Multiple Choice

A velocity vector of magnitude 1.2m/s is horizontal. A second velocity vector of magnitude 2.0 m/s must be added to the first so that the sum is vertical in direction. Find the direction of the second vector and the magnitude of the sum of the two vectors

• 37⁰

  1.6 m/s

• 74⁰

  1.6 m/s

• 37⁰

  3.2 m/s

• 74⁰

  3.2 m/s

Components of a Vector

• When resolving vectors in two directions, vectors can be resolved into a pair of perpendicular components.

• Components are defined along an axis (x,y) 

  • The x component of a vector is defined as:

    • M_x = McosΘ

  • The y component of a vector is defined as:

    • M_y = MsinΘ

Multiple Choice

Find the components of the vectors in the figure below. The magnitude of a is 12.0 units and that of b is 24 units

• a_x= -4.24

  a_y= -4.24

  b_x= 12.0

  b_y= -20.8

• a_x= -4.24

  a_y= -4.24

  b_x= 20.8

  b_y= -12.0

• a_x= 8.49

  a_y= 8.49

  b_x= 12.0

  b_y= -20.8

• a_x= -8.49

  a_y= -8.49

  b_x= 20.8

  b_y= -12.0

​Reconstructing a Vector from its Components

Nature of Science

• The idea of using distance and direction to mark and navigate across the world has been around for thousands of years.

• The concept of vectors and the algebra used to manipulate them were introduced in the first half of the 19th century to represent real and complex numbers

• In this section you’ve worked on vectors in two dimensions but vector algebra can address two three or multiple dimensional planes.