DisplacementDisplacement

Defining PositionDefining Position

Position has three properties:Position has three properties:• Origin, magnitude, directionOrigin, magnitude, direction

1 dimension1 dimension12 feet above sea 12 feet above sea

level.level.

Origin: sea levelOrigin: sea level

Magnitude: 12 Magnitude: 12 feetfeet

Direction: upDirection: up

2 dimensions2 dimensions65 miles west of 65 miles west of

Chicago.Chicago.

Origin: Origin: downtown downtown ChicagoChicago

Magnitude: 65 Magnitude: 65 milesmiles

Direction: westDirection: west

3 dimensions3 dimensionsRange 200 m, Range 200 m,

bearing 270bearing 270, , at 30at 30 altitude. altitude.

Origin: observerOrigin: observer Magnitude: 200 Magnitude: 200

meters meters Direction: 270Direction: 270

by the compass by the compass and 30and 30 up. up.

Position GraphPosition Graph

Position can be displayed on Position can be displayed on a graph.a graph.• The origin for position is the The origin for position is the

origin on the graph.origin on the graph.

• Axes are position Axes are position coordinates.coordinates.

• The position is a vector.The position is a vector.

A set of position points A set of position points connected on a graph is a connected on a graph is a trajectorytrajectory..

y

x

r

2-dimensions (x, y)

position vector

trajectory

Scalar MultiplicationScalar Multiplication

A vector can be multiplied by A vector can be multiplied by a scalar.a scalar.• Change feet to meters.Change feet to meters.

• Walk twice as far in the Walk twice as far in the same direction.same direction.

Scalar multiplication Scalar multiplication multiplies each component multiplies each component by the same factor.by the same factor.

The result is a new vector, The result is a new vector, always parallel to the original always parallel to the original vector.vector.

),( yx sAsAAsT

Reference PointReference Point

Displacement is different Displacement is different from positionfrom position

• PositionPosition is measured is measured relative to an origin common relative to an origin common to all points.to all points.

• DisplacementDisplacement is measured is measured relative to the object’s initial relative to the object’s initial position.position.

• The path (trajectory) doesn’t The path (trajectory) doesn’t matter for displacement.matter for displacement.

trajectory displacement

originposition

Displacement VectorDisplacement Vector

The position vector is often The position vector is often designated by .designated by .

A change in a quantity is A change in a quantity is designated by designated by ΔΔ ( (deltadelta).).

Always take the final value Always take the final value and subtract the initial value.and subtract the initial value.

y

x2r1r

12 rrr

r

Two DisplacementsTwo Displacements

A hiker starts at a point 2.0 km east of camp, then A hiker starts at a point 2.0 km east of camp, then walks to a point 3.0 km northeast of camp. What is walks to a point 3.0 km northeast of camp. What is the displacement of the hiker?the displacement of the hiker?

Each individual displacement is a vector that can be Each individual displacement is a vector that can be represented by an arrow.represented by an arrow.

2.0 km

3.0 km

Vector SubtractionVector Subtraction

To subtract two vectors, place both at the same To subtract two vectors, place both at the same origin.origin.

Start at the tip of the first and go to the tip of the Start at the tip of the first and go to the tip of the second.second.

ABD

B

A

D

Component SubtractionComponent Subtraction

Multiplying a vector by Multiplying a vector by 1 will create an antiparallel 1 will create an antiparallel vector of the same magnitude.vector of the same magnitude.

Vector subtraction is equivalent to scalar Vector subtraction is equivalent to scalar multiplication and addition.multiplication and addition.

yyyyy

xxxxx

ABABD

ABABD

ABABD

)1(

)1(

)1(

ABD

)1(

B

A

D

Displacement ComponentsDisplacement Components

ABD

Find the components of each Find the components of each vector, and subtract.vector, and subtract.

• AAxx = 2.0 km = 2.0 km

• AAyy = 0.0 km = 0.0 km

• BBxx = (3.0 km)cos45 = 2.1 km = (3.0 km)cos45 = 2.1 km

• BByy = (3.0 km)sin45 = 2.1 km = (3.0 km)sin45 = 2.1 km

• DDxx = B = Bxx – A – Axx = 0.1 km = 0.1 km

• DDyy = B = Byy – A – Ayy = 2.1 km = 2.1 km

B

A

D

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