VectorsVectors

Measured quantity with Measured quantity with MagnitudeMagnitude and and DirectionDirection..

Example:Example: The wind velocity of The wind velocity of 30 knots 30 knots NorthNorth The weight of The weight of 140 lbs. 140 lbs. downdown A displacement of A displacement of 5m 5m WestWest

Vector NotationVector Notation

VectorVector::

vv or or

Scalar:Scalar:

V or V or

v

v

Vector AdditionVector Addition

Parallel vectors behave like numbers Parallel vectors behave like numbers on a number line. on a number line.

AddAdd the magnitudes of vectors in the the magnitudes of vectors in the samesame direction. direction.

SubtractSubtract the magnitudes of vectors the magnitudes of vectors in in oppositeopposite directions. directions.

Graphical AdditionGraphical Addition

Vectors can be added with scaled Vectors can be added with scaled drawings. Note that vector addition is drawings. Note that vector addition is commutative.commutative.

Add2Vectors.htmlAdd2Vectors.htmlAdd3Vectors.htmlAdd3Vectors.html

The sum of two or more vectors is a The sum of two or more vectors is a ResultantResultant

Vector ComponentsVector Components

In some cases it is easiest way to In some cases it is easiest way to combine vectors is with components. A combine vectors is with components. A Vector ComponentVector Component is the portion of is the portion of the vector that lies along an x or y axis. the vector that lies along an x or y axis. We use trigonometry to find these We use trigonometry to find these components.components.

phetphet component addition component addition

U of T add componentsU of T add components

Unit VectorsUnit Vectors

All vectors have direction. In many All vectors have direction. In many cases it is helpful to define a direction cases it is helpful to define a direction as along an axis. Vectors that perform as along an axis. Vectors that perform this service are called this service are called Unit VectorsUnit Vectors and are used in component notation. and are used in component notation. Unit vectors are orthogonal or Unit vectors are orthogonal or mutually perpendicular.mutually perpendicular.

UnitVectors.htmlUnitVectors.html

Vector MultiplicationVector Multiplication

There are two ways to multiply There are two ways to multiply vectors. They are not vectors. They are not interchangeable!interchangeable!

The equation used will determine the The equation used will determine the nature of the productnature of the product

Scalar ProductScalar Product

The Scalar or The Scalar or Dot ProductDot Product is used to is used to multiply only the portion of vectors multiply only the portion of vectors that are that are parallelparallel. .

DotProduct.htmlDotProduct.html

cosABBA

Right Hand RuleRight Hand Rule

Physics Gang SignPhysics Gang Sign

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Vector ProductVector ProductThe Vector or The Vector or Cross ProductCross Product is used to is used to

multiply vectors and get a vector answer. multiply vectors and get a vector answer. While it is defined as:While it is defined as:

There is a more compact way to get an There is a more compact way to get an answer with components. The product is answer with components. The product is always a vector and always orthogonal to always a vector and always orthogonal to AA and and BB..

CrossProduct.htmlCrossProduct.html

RightHandRule.htmlRightHandRule.html

sinABBA

The Vector product is Anti-The Vector product is Anti-Commutative. This means Commutative. This means A A x x BB==-B -B

x x AA

RightHandRule.htmlRightHandRule.html

Find Vector Product with Find Vector Product with DeterminatesDeterminates

Matrix algebra provides a compact method Matrix algebra provides a compact method for writing and solving cross products. For for writing and solving cross products. For two vectors two vectors AA=2=2ii+3+3jj+5+5kk and and BB=4=4ii--33jj+2+2k.k.

We can write We can write AAxxBB as: as:

and calculate the determinant as:and calculate the determinant as: ii[(3*2)-(5*-3)]-[(3*2)-(5*-3)]-jj[(2*2)-(5*4)]+[(2*2)-(5*4)]+kk[(2*-3)-[(2*-3)-

(3*4)] =21(3*4)] =21ii+5+5jj-6-6kk

234

532

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