An Vectors 2: Algebra of Vectors Department of Mathematics University of Leicester

Contents Introduction Magnitude Vector Addition Scalar Multiplication

Scalar Multiplication Introduction Magnitude Vector Addition Scalar Multiplication 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. Magnitude = length x y v Next

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

y x

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

Magnitude of a Vector – 3 Dimensions Introduction Magnitude Vector Addition Scalar Multiplication 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 y b x a Next

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

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

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

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

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

y x y This is blank, ready to go into next slide. x

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

OR: y Create the parallelogram Draw the 1st vector a+b a Draw the 2nd vector b x Repeat Draw the 2nd vector STARTING FROM the first vector y b This slide doesn’t do anything – it’s just there to go straight into next slide – to REPEAT TOP RIGHT GRAPH Back to Vector Addition Draw the 1st vector The end is the new vector a a+b Repeat x

OR: y Create the parallelogram Draw the 1st vector a+b a Draw the 2nd vector b x Repeat Draw the 2nd vector STARTING FROM the first vector y b This just animates top-right graph Back to Vector Addition Draw the 1st vector The end is the new vector a a+b Repeat x

OR: y Create the parallelogram Draw the 1st vector a+b a Draw the 2nd vector b x Repeat Draw the 2nd vector STARTING FROM the first vector y b This slide doesn’t do anything – it’s just there to go straight into next slide – to REPEAT BOTTOM LEFT GRAPH Back to Vector Addition Draw the 1st vector The end is the new vector a a+b Repeat x

OR: y Create the parallelogram Draw the 1st vector a+b a Draw the 2nd vector b x Repeat Draw the 2nd vector STARTING FROM the first vector y b This just animates bottom-left graph Back to Vector Addition Draw the 1st vector The end is the new vector a a+b Repeat x

Scalar Multiplication Introduction Magnitude Vector Addition Scalar Multiplication 8 6 + + 4 2 -8 -6 -4 -2 2 4 6 8 -2 Origin is (210,320) -4 Blue are the vectors, Pink is the sum. -6 -8 Next

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

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

Try multiplying some vectors by scalars Introduction Magnitude Vector Addition Scalar Multiplication Try multiplying some vectors by scalars 8 6 4 2 v is pink λv blue -8 -6 -4 -2 2 4 6 8 Origin is (210,320) -2 -4 -6 -8 Next

Scalar Multiplication Introduction Magnitude Vector Addition Scalar Multiplication Questions… What is ?

Scalar Multiplication Introduction Magnitude Vector Addition Scalar Multiplication Questions… What is ?

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

Scalar Multiplication Introduction Magnitude Vector Addition Scalar Multiplication 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). Next