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

2. Contents
Introduction
Magnitude
Vector Addition
Scalar Multiplication

3. 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
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4. 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
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5. y x

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

7. 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
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8. 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
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9. Scalar Multiplication
Introduction
Magnitude
Vector Addition
Scalar Multiplication
Questions…
What is the magnitude of ?

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

11. 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
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12. 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
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13. y x y
This is blank, ready to go into next slide. x

14. 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

15. 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

16. 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

17. 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

18. 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

19. 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
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20. 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.
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21. Scalar Multiplication- Geometry
Introduction
Magnitude
Vector Addition
Scalar Multiplication
Scalar Multiplication- Geometry y 2a a x
(-1)a
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22. 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
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23. Scalar Multiplication
Introduction
Magnitude
Vector Addition
Scalar Multiplication
Questions…
What is ?

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

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

26. 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).
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