Introduction to VectorsIntroduction to Vectors

Chapter 3 section 1Chapter 3 section 1

Scalar QuantityScalar Quantity

ScalarScalar – A quantity that can be – A quantity that can be completely specified by its completely specified by its magnitude, but magnitude, but NONO direction. direction. Examples:Examples:

SpeedSpeed DistanceDistance VolumeVolume EnergyEnergy TimeTime MassMass

Vector QuantityVector Quantity

VectorVector – A quantity that can be – A quantity that can be described by its magnitude described by its magnitude ANDAND its its direction.direction. Examples:Examples:

DisplacementDisplacement VelocityVelocity AccelerationAcceleration ForceForce MomentumMomentum

Vectors and Scalars in the BookVectors and Scalars in the Book

Vectors quantities are indicated in Vectors quantities are indicated in BoldfaceBoldface by their variable. by their variable. v = v = 3030 m/sm/s

Scalars quantities are indicated in Scalars quantities are indicated in italicsitalics by their variable. by their variable. t = t = 3.23.2 ss

Vector DiagramsVector Diagrams

In diagrams, vectors are shown as In diagrams, vectors are shown as arrows that point the direction of its arrows that point the direction of its magnitude.magnitude. Length of arrow: Magnitude of the Length of arrow: Magnitude of the

vectorvector Direction of arrow: Path of vectorDirection of arrow: Path of vectorv = 30 mi/hr

v = 60 mi/hr

ResultantResultant

ResultantResultant – A vector representing the – A vector representing the sum of two or more vectors.sum of two or more vectors.

When adding vectors, they must When adding vectors, they must have the same units and describe have the same units and describe similar quantities.similar quantities. Example:Example:

The sum of the vectors must all be velocity The sum of the vectors must all be velocity and must be in units of m/s.and must be in units of m/s.

Vector Addition in 1-DimensionVector Addition in 1-Dimension

A=5m B=2m R=7m

A+B=R5m + 2m =

7m

A+B=R5m + (-3m) =

2m

A=5m B=3m R=2m

Head-to-Tail MethodHead-to-Tail Method

To add two (or more) vectors together To add two (or more) vectors together graphically using the head-to-tail method graphically using the head-to-tail method you simply draw the first vector and then you simply draw the first vector and then draw the second vector with its tail at the draw the second vector with its tail at the head of the first vector.head of the first vector.

If there are more vectors to be added draw If there are more vectors to be added draw each one with its tail at the head of the each one with its tail at the head of the preceding one. The sum or resultant is a preceding one. The sum or resultant is a vector drawn from the tail of the first vector drawn from the tail of the first vector to the head of the last vector. It vector to the head of the last vector. It does not matter in which order you add does not matter in which order you add them.them.

VectorsVectors

HEADTAIL

The resultant always measures from where

you started to where you end at.

Cartesian PlaneCartesian Plane

0°

270°

90°

180°

Vector AnglesVector Angles

When describing the direction of a vector, When describing the direction of a vector, the angle always starts at the 0 degrees on the angle always starts at the 0 degrees on the x-axis and moves counter-clockwise to the x-axis and moves counter-clockwise to reference the direction of the vector.reference the direction of the vector.

θ=45° θ=105°

θ=300°

Properties of VectorsProperties of Vectors

1.1. Vectors can be moved parallel to Vectors can be moved parallel to themselves in a diagram, as long as themselves in a diagram, as long as the magnitude stays the same.the magnitude stays the same.

The horizontal vector is moved and doesn’t change the problem The horizontal vector is moved and doesn’t change the problem as long as the magnitude doesn’t change and it remains parallel as long as the magnitude doesn’t change and it remains parallel to its original position.to its original position.

Properties of VectorsProperties of Vectors

2. Vectors can be added in any direction.2. Vectors can be added in any direction.

Ex:Ex: A+B+C+D+E= ResultantA+B+C+D+E= Resultant

B+A+D+C+E= ResultantB+A+D+C+E= Resultant

Resultant

Resultant

AB

E

D

CB

A

E

DC

Properties of VectorsProperties of Vectors

3. To subtract a vector, add its 3. To subtract a vector, add its oppositeopposite

A - B = A + (-B)A - B = A + (-B)

Properties of VectorsProperties of Vectors

4. Multiplying or dividing vectors by 4. Multiplying or dividing vectors by scalars results in vectors.scalars results in vectors.

3 3 • A = 3A• A = 3A