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www.le.ac.uk An Vectors 2: Algebra of Vectors Department of Mathematics University of Leicester

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Contents Introduction Magnitude Vector Addition Scalar Multiplication

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Introduction A vector has size and direction. The size or magnitude of a vector means the length from its start point to its end point. You can add vectors together, and also multiply them by scalars. Introduction Magnitude Vector Addition Scalar Multiplication Next x y v Magnitude = length

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Take the vector Then its magnitude is found by Pythagoras’s Theorem: If its magnitude is 1, it is a unit vector Magnitude of a Vector Introduction Magnitude Vector Addition Scalar Multiplication Click here to see a proof Next

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x y

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x y Click here to repeat Click here to go back (by Pythagoras’s Theorem)

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Magnitude of a Vector – 3 Dimensions Consider the vector (a, b, c) in 3D and it’s projection onto the x-y plane. By Pythagoras’ Theorem we know the magnitude of this is x y a b Introduction Magnitude Vector Addition Scalar Multiplication Next

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Now consider how far it goes up in the z direction Magnitude of a Vector – 3 Dimensions z We know the length of this is. We know the length of this is c from the origin Again, by Pythagoras: Introduction Magnitude Vector Addition Scalar Multiplication x-y plane Next

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Questions… What is the magnitude of ? Introduction Magnitude Vector Addition Scalar Multiplication

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Questions… What is the magnitude of this vector: ? x y 2 4 Introduction Magnitude Vector Addition Scalar Multiplication

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Vector Addition To add vectors together, we add together the elements of the same rows Take the vectors and Introduction Magnitude Vector Addition Scalar Multiplication Next

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Vector Addition- Geometry b a a+ba+b Click here to see how the vectors add together x y Introduction Magnitude Vector Addition Scalar Multiplication Next

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x x y y

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x y b a x y a a+ba+b b OR: Create the parallelogram Draw the 2 nd vector Draw the 1 st vector Draw the 2 nd vector STARTING FROM the first vector The end is the new vector Back to Vector Addition Repeat a+ba+b

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x y b a x y a a+ba+b b OR: Create the parallelogram Draw the 2 nd vector Draw the 1 st vector Draw the 2 nd vector STARTING FROM the first vector The end is the new vector Back to Vector Addition Repeat a+ba+b

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x y b a x y a a+ba+b b OR: Create the parallelogram Draw the 2 nd vector Draw the 1 st vector Draw the 2 nd vector STARTING FROM the first vector The end is the new vector Back to Vector Addition Repeat a+ba+b

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x y b a x y a a+ba+b b OR: Create the parallelogram Draw the 2 nd vector Draw the 1 st vector Draw the 2 nd vector STARTING FROM the first vector The end is the new vector Back to Vector Addition Repeat a+ba+b

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x y b a x y a a+ba+b b OR: Create the parallelogram Draw the 2 nd vector Draw the 1 st vector Draw the 2 nd vector STARTING FROM the first vector The end is the new vector Back to Vector Addition Repeat a+ba+b

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+ 2 -2 4 6 8 -6 -8 -4 0 -2 -6 -4 24 68 -8 Introduction Magnitude Vector Addition Scalar Multiplication Next Blue are the vectors, Pink is the sum.

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Scalar Multiplication To multiply by a scalar, we just multiply each part of the vector by the scalar, individually. Introduction Magnitude Vector Addition Scalar Multiplication Next

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Scalar Multiplication- Geometry x y a 2a2a (-1)a Introduction Magnitude Vector Addition Scalar Multiplication Next

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Try multiplying some vectors by scalars 2 -2 4 6 8 -6 -8 -4 0 -2 -6 -4 24 68 -8 v is pink λv blue Introduction Magnitude Vector Addition Scalar Multiplication Next

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Questions… What is ? Introduction Magnitude Vector Addition Scalar Multiplication

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Questions… What is ? Introduction Magnitude Vector Addition Scalar Multiplication

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Questions… Which of these vectors is 3a + b? x y x y a b Introduction Magnitude Vector Addition Scalar Multiplication

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Conclusion The magnitude of a vector can be found using Pythagoras’s theorem. This can be extended to any number of dimensions. Vectors can be added and multiplied by scalars. (You can’t multiply two vectors together). Introduction Magnitude Vector Addition Scalar Multiplication Next

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