Using VectorsUsing Vectors

Two DisplacementsTwo Displacements

A hiker walks east from camp for 2.0 km, then A hiker walks east from camp for 2.0 km, then northeast for 3.0 km. What is the final displacement northeast for 3.0 km. What is the final displacement of the hiker?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

Graphical AdditionGraphical Addition

The two vectors can be added The two vectors can be added graphically.graphically.

The tail of the second vector is The tail of the second vector is placed at the tip of the first.placed at the tip of the first.

The length and directions are The length and directions are kept the same.kept the same.

The result is the total The result is the total displacement. It can be displacement. It can be measured directly.measured directly.

2.0 km

3.0 km

4.6 km

CBA

Commutative PropertyCommutative Property

Vectors can be shifted as long as they don’t change Vectors can be shifted as long as they don’t change direction and magnitude.direction and magnitude.

Vectors can be added in reverse order and get the Vectors can be added in reverse order and get the same result.same result.

CAB

CBA

ParallelogramParallelogram

If two vectors are added from a common origin one If two vectors are added from a common origin one can be shifted to make a parallelogram.can be shifted to make a parallelogram.

This is the same as putting the tail to the tip.This is the same as putting the tail to the tip.

CBA

Parallel and AntiparallelParallel and Antiparallel

Vectors that point in the Vectors that point in the same direction are same direction are parallelparallel..

Vectors that point in opposite Vectors that point in opposite directions are directions are antiparallelantiparallel..

CancellationCancellation

What happens if we add two antiparallel vectors of What happens if we add two antiparallel vectors of equal magnitude?equal magnitude?

The vector sum is a zero length vector. The vectors The vector sum is a zero length vector. The vectors cancel out.cancel out.

A

A

0)( AA

Vector SubtractionVector Subtraction

To subtract one vector, add the antiparallel vector To subtract one vector, add the antiparallel vector instead.instead.

DBA

Vector TrianglesVector Triangles

Three vectors have the Three vectors have the same magnitude, L, and same magnitude, L, and form an equilateral triangle.form an equilateral triangle.

Find Find • They are already tip to tail, They are already tip to tail,

soso

Find Find • Redraw the pictureRedraw the picture

• The direction is upThe direction is up

• The magnitude is The magnitude is

A

B

C

BA

CBA

BA

60sin260sin60sin LLL

Lsin6060o

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