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Vectors: 5 Minute Review Vectors can be added or subtracted. ◦ To add vectors graphically, draw one after the other, tip to tail. ◦ To add vectors algebraically, Resolve the vectors into components. Add the components of each direction. Use the Pythagorean Theorem to find the magnitude of the resultant vector and inverse tan to find the angle.

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Vector Addition Example A boy pushes horizontally on a wagon with a force of 10 N. A girl pulls on the wagon’s handle at an angle of 30 degrees from the horizontal with a force of 8 N. What is the net force acting on the wagon?

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Unit Vectors A unit vector is a vector that points along the x, y or z axis and is one unit long. ◦ The symbols for the x, y, and z unit vectors are ◦ Any vector can be expressed as a sum of its x, y and z components multiplied by unit vectors. ◦ The dimensions of the quantity are stated along with the unit vectors, e.g.

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©2008 by W.H. Freeman and Company

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Unit Vector Example A car has a velocity of 6 m/s in a direction 30 o north of east. Express this vector in terms of unit vectors. Let east be the positive x direction and north the positive y direction.

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Operations with Vectors: Magnitude To find the magnitude of a vector, use the Pythagorean Theorem.

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Operations with Vectors: Addition and subtraction To add two vectors, add their components.

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Vector Addition & Subtracting Graphical method Addition: Connect head to tail Subtraction: flip the subtrahend 180°, then connect head to tail.

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Multiplication of Vectors: 1.Vector Product of a Scalar and a Vector 2.Scalar Product of Two Vectors 3.Vector Product of Two Vectors

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Multiplication of Vector by Scalar Applications momentump = mv electric forceF = qE Result A vector with the same direction, a different magnitude and perhaps different units.

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Multiplication of Vector by Vector (Dot Product) Application workW = F d Result A scalar with magnitude and no direction.

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Multiplication of Vector by Vector (Dot Product) C = A B C = AB cos A B

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Dot Product Practice i i = i j = i k = j j = j k = j i = k i = k k = k j = 1x1x cos0 1x1x cos 90 5i 3i = 4i 2j = -4i 2k = 5j -3j = 2j 2 k = 1j 9 i = 9k 7 i = -√3k -2k = ½k √2j =

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Multiplication of Vector by Vector (Cross Product) Application Work = r F Magnetic force F = qv B Result A vector with magnitude and a direction perpendicular to the plane established by the other two vectors.

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Multiplication of Vector by Vector (Cross Product) C = A B C = AB sin (magnitude) Direction determined by Right Hand Rule A B

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Multiplication of Vector by Vector (Cross Product) C = A B A B

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Operations with Vectors: The Cross Product The cross product yields a vector at right angles to the two vectors.

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Cross Product Practice i x i = i x j = i x k = j x j = j x k = j x i = k x i = k x k = k x j = 1x1x sin0 1x1x sin90 1x1x sin0 1x1x sin90 5i x4i = 3i x 3j = -9i x 6k = 5j x5 j = √3j x √3k = 2.4j x 2 i = ½k x ¾i = -2k x 2k = -2k x 2j =

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Cross Product Practice i x i = i x j = i x k = j x j = j x k = j x i = k x i = k x k = k x j = 1x1x sin0 1x1x sin90 1x1x sin0 1x1x sin90 5i x4i = 3i x 3j = -9i x 6k = 5j x5 j = √3j x √3k = 2.4j x 2 i = ½k x ¾i = -2k x 2k = -2k x 2j =

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Vector Multiplication: 2 types: Cross product & dot product

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