1. Two-Dimensional Motion and Vectors Introduction to Vectors
Chapter 3: Section 1

2. Learning Targets Distinguish between a scalar and a vector
Add and subtract vectors by using the graphical method
Multiply and divide vectors and scalars
P3.2A, P3.2C

3. Reviewing Scalars and Vectors
Vectors indicate direction while scalars do not
A scalar is a quantity that has magnitude but no direction
speed, volume, the number of pages in a textbook
A vector is a physical quantity that has both direction and magnitude
displacement, velocity, acceleration

4. Representing Vectors and Scalars
Vectors are represented by boldface symbols while scalar quantities are represented by italics
Velocity, which includes direction, is written v = 3.5 m/s to the north
The speed of a bird is written v = 3.5 m/s
You can also represent a vector by writing an arrow above the abbreviation for the quantity
v = 3.5 m/s to the north

5. Vector Diagrams
Vector quantities are often represented by scaled vector diagrams.
In vector diagrams, vectors are shown as arrows that point in the direction of the vector.
The length of the vector arrow is proportional to the vector’s magnitude

6. Characteristics of Vector Diagrams
A scale is clearly listed
A vector arrow is drawn in a specified direction.
The vector arrow has a head and a tail.
The magnitude and direction of the vector is clearly labeled.

7. Direction of Vectors
The direction of a vector is often expressed as an angle of rotation of the vector about its “tail".
For example, a vector can be said to have a direction of 40 degrees North of East
This means the vector pointing East has been rotated 40 degrees towards the northerly direction

8. The direction of a vector can also be expressed as a counterclockwise angle of rotation of the vector about its “tail" from due East.
A vector said to have a direction of 240 means that if the tail of the vector was pinned down, the vector was rotated an angle of 240 in the counterclockwise direction beginning from due east.

9. Magnitude of a Vector
The magnitude of a vector in a scaled vector diagram is depicted by the length of the arrow.

10. Resultant Vectors
When adding vectors you must make certain that they have the same units and describe similar quantities.
Can’t add a velocity vector to a displacement vector
Can’t add meters and feet
The answer found by adding two vectors is called the resultant

11. Properties of Vectors 1. Vectors can be added graphically
Imagine a toy car that is moving
0.8 m/s across a walkway that
moves at 1.5 m/s forward
The car’s resultant velocity while
moving from one side of the
walkway to the other will be the
combination of two independent
motions

12. The car’s resultant vector can be drawn from the tail of the first vector to the tip of the last vector.
The magnitude can be measured
using a ruler and the direction
with a protractor

13. 2. Vectors can be added in any order
When two or more vectors are added, the sum is independent of the order of the addition
Picture a runner training for a marathon along city streets
In the picture below, the runner executes the same four displacements, but in a different order
Regardless of the path, the total displacement is the same

14. 3. To subtract a vector, add its opposite
The negative of a vector is defined as a vector with the same magnitude as the original vector but opposite in direction.
The negative velocity of a car traveling 30 m/s to the west is -30 m/s to the west, or +30 m/s to the east
Adding a vector to its negative vector gives zero

15. 4. Multiplying or dividing vectors by scalars results in vectors
If a cab driver drives twice as fast, the cab’s original velocity vector (vcab) is multiplied by the scalar number 2.
The result, written 2vcab is a vector with a magnitude twice that of the original vector and pointing in the same direction
Going twice as fast in the opposite direction would be -2vcab