vectors measured quantity with magnitude and direction. example: the wind velocity of 30 knots north...
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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
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Vector NotationVector Notation
VectorVector::
vv or or
Scalar:Scalar:
V or V or
v
v
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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.
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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
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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
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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
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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
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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
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Right Hand RuleRight Hand Rule
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Physics Gang SignPhysics Gang Sign
tshirt
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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
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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
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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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kji