# Calc10_2 - QuickTime and a decompressor are needed to see...

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QuickTimeª and a decompressor are needed to see this picture. 10.2 Vectors in the Plane

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Warning: Only some of this is review .
Quantities that we measure that have magnitude but not direction are called scalars . Quantities such as force, displacement or velocity that have direction as well as magnitude are represented by directed line segments . A B initial point terminal point AB uuu v The length is AB uuu v

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A B initial point terminal point AB uuu v A vector is represented by a directed line segment. Vectors are equal if they have the same length and direction (same slope).
A vector is in standard position if the initial point is at the origin. x y ( 29 1 2 , v v The component form of this vector is: 1 2 , v v = ω

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A vector is in standard position if the initial point is at the origin. x y ( 29 1 2 , v v The component form of this vector is: 1 2 , v v = ω The magnitude (length) of 1 2 , v v = ω is: 2 2 1 2 v v = + ω
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 P Q (-3,4) (-5,2) The component form of PQ uuu v is: 2, 2 = - - ω v (-2,-2) ( 29 ( 29 2 2 2 2 = - + - v 8 = 2 2 =

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If 1 = ω Then v is a unit vector . 0,0 is the zero vector and has no direction.
Vector Operations: 1 2 1 2 Let , , , , a scalar (real number). u u v v k = = υ ω 1 2 1 2 1 1 2 2 , , , u u v v u v u v + = + = + + υ ω (Add the components.) 1 2 1 2 1 1 2 2 , , , u u v v u v u v - = - = - - υ ω (Subtract the components.)

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Vector Operations: Scalar Multiplication: 1 2 , k ku ku = υ Negative (opposite): ( 29 1 2 1 , u u - = - = - - u u
v v u u u+v u + v is the resultant vector . (Parallelogram law of addition)

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The angle between two vectors is given by: 1 1 1 2 2 cos u v u v θ - + = υ ω This comes from the law of cosines.
The dot product (also called inner product ) is defined as: 1 1 2 2 cos u v u v θ × = = + υ ω υ ω Read “u dot v” Example: 3,4 5,2 × ( 29 ( 29 ( 29 ( 29 3 5 4 2 = + 23 =

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The dot product (also called inner product ) is defined as: 1 1 2 2 cos u v u v θ × = = + υ ω υ ω This could be substituted in the formula for the angle between vectors (or solved for theta) to give: 1 cos - × = ÷ ÷ υ ω υ ω
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Calc10_2 - QuickTime and a decompressor are needed to see...

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