lesson #5 continuation

Scalars vs Vectors

Multiple Choice

A scalar quantity has . . .

• magnitude only

• direction only

• both magnitude and direction

• neither

Multiple Choice

A vector quantity has . . .

• magnitude only

• direction only

• both magnitude and direction

• neither

Multiple Choice

Which quantity do we use an arrow to represent?

• scalars

• vectors

• The tail is the starting position.

• The head is the ending position.

• The magnitude represents how long or large the vector is.

  • For example, a vector representing 100 should be twice as long as a vector representing 50. ​

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​How To Draw a Vector

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Match

Label each as a "scalar" or a "vector"

• scalar (kg)

• scalar (s)

• scalar (kg/m^3)

• vector (N)

• vector (m/s)

mass

time

density

force

velocity

• mass

• time

• density

• force

• velocity

• 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

Scaling Vectors

Multiple Choice

Given $$\overrightarrow{C}$$ = <-4, 5> find 2 $$\overrightarrow{C}$$

• <-12, 4>

• <-2, 3>

• <-8, 10>

• <-2, 2.5>

Multiple Choice

If $$\overrightarrow{A}$$ = <-6, 2> and $$\overrightarrow{B}$$ = <1, 7>, find $$\overrightarrow{A}$$ + $$\overrightarrow{B}$$

• <-7, -5>

• <7, 9>

• <-5, 9>

• <-4, 8>

Multiple Choice

If $$\overrightarrow{a}$$ = <-6, 2> and $$\overrightarrow{b}$$ = <1, 7> and $$\overrightarrow{c}$$ = <-4, 5>, find $$\overrightarrow{a}$$ - 3 $$\overrightarrow{c}$$

• <-18, 17>

• <-2, -3>

• <6, -13>

• <5, -2>

Components of a Vector

• There will be times when it's best to treat vectors algebraically, so we create a coordinate system similar to what we've used in the past for a point in 2 or 3 dimensions

• A two-dimension coordinate is written as (x, y). A vector will be written as .

• The position vector is the vector that begins at the origin and goes to the point.

EVERY vector has x- and y-components.

A_x = A*cos(θ)

Ay = A*sin(θ)

Multiple Choice

If v =〈a,b〉, then a is the ___________ component of v.

• vertical

• horizontal

• parallel

• slope

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

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