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![Page 1: Www.le.ac.uk Vectors 1: An Introduction to Vectors Department of Mathematics University of Leicester.](https://reader036.fdocuments.us/reader036/viewer/2022081507/5519b0485503465b578b4574/html5/thumbnails/1.jpg)
www.le.ac.uk
Vectors 1:An Introduction to Vectors
Department of MathematicsUniversity of Leicester
![Page 2: Www.le.ac.uk Vectors 1: An Introduction to Vectors Department of Mathematics University of Leicester.](https://reader036.fdocuments.us/reader036/viewer/2022081507/5519b0485503465b578b4574/html5/thumbnails/2.jpg)
Contents
What is a Vector?
Dimensions
Basis
Drawing Vectors
![Page 3: Www.le.ac.uk Vectors 1: An Introduction to Vectors Department of Mathematics University of Leicester.](https://reader036.fdocuments.us/reader036/viewer/2022081507/5519b0485503465b578b4574/html5/thumbnails/3.jpg)
What is a vector?
• A vector is a quantity which has direction as well as length
• They are used to determine the position of one point in space relative to another
What is a Vector? Dimensions Basis
Drawing Vectors
Next
![Page 4: Www.le.ac.uk Vectors 1: An Introduction to Vectors Department of Mathematics University of Leicester.](https://reader036.fdocuments.us/reader036/viewer/2022081507/5519b0485503465b578b4574/html5/thumbnails/4.jpg)
Spot the DifferenceSo, what is the difference between a vector
and a scalar?
Next
What is a Vector? Dimensions Basis
Drawing Vectors
![Page 5: Www.le.ac.uk Vectors 1: An Introduction to Vectors Department of Mathematics University of Leicester.](https://reader036.fdocuments.us/reader036/viewer/2022081507/5519b0485503465b578b4574/html5/thumbnails/5.jpg)
Spot the DifferenceSo, what is the difference between a vector
and a scalar?
• A scalar only has magnitude and is independent of direction
My car goes at 40mph.Scalar
man
Next
What is a Vector? Dimensions Basis
Drawing Vectors
![Page 6: Www.le.ac.uk Vectors 1: An Introduction to Vectors Department of Mathematics University of Leicester.](https://reader036.fdocuments.us/reader036/viewer/2022081507/5519b0485503465b578b4574/html5/thumbnails/6.jpg)
Spot the DifferenceSo, what is the difference between a vector
and a scalar?
• A scalar only has magnitude and is independent of direction
• A vector has both magnitude AND direction
I’m going 20 miles West.
Vector man
My car goes at 40mph.Scalar
man
Next
What is a Vector? Dimensions Basis
Drawing Vectors
![Page 7: Www.le.ac.uk Vectors 1: An Introduction to Vectors Department of Mathematics University of Leicester.](https://reader036.fdocuments.us/reader036/viewer/2022081507/5519b0485503465b578b4574/html5/thumbnails/7.jpg)
Spot the Difference
• Do we use a scalar or vector quantity to represent the top speed of a car?
Next
What is a Vector? Dimensions Basis
Drawing Vectors
![Page 8: Www.le.ac.uk Vectors 1: An Introduction to Vectors Department of Mathematics University of Leicester.](https://reader036.fdocuments.us/reader036/viewer/2022081507/5519b0485503465b578b4574/html5/thumbnails/8.jpg)
Spot the Difference
• Do we use a scalar or vector quantity to represent the top speed of a car?
A: We use a SCALAR ie. the top speed of my car is 120mph. However, if I were to talk about the velocity of the car, this concerns both the speed AND direction in which I am travelling, so a vector is used for velocity.
Next
What is a Vector? Dimensions Basis
Drawing Vectors
![Page 9: Www.le.ac.uk Vectors 1: An Introduction to Vectors Department of Mathematics University of Leicester.](https://reader036.fdocuments.us/reader036/viewer/2022081507/5519b0485503465b578b4574/html5/thumbnails/9.jpg)
Question…Choose whether the following would be measured using vectors or scalars:
Temperature
Position of the Earth in it’s orbit around the sun.
Distance travelled on a circular walk.
Next
What is a Vector? Dimensions Basis
Drawing Vectors
ScalarVector
ScalarVector
ScalarVector
Check Answers Clear Answers Show Answers
![Page 10: Www.le.ac.uk Vectors 1: An Introduction to Vectors Department of Mathematics University of Leicester.](https://reader036.fdocuments.us/reader036/viewer/2022081507/5519b0485503465b578b4574/html5/thumbnails/10.jpg)
Dimensions
• The formal definition of dimension is the minimum number of coordinates used to specify a point in that dimension.
• A LINE is one dimensional; we only need a single number to get to any point along it (think about the number line).
• A PLANE or SURFACE is two dimensional; just think about a graph. We need the value along the x and y axis to find a point. Next
What is a Vector? Dimensions Basis
Drawing Vectors
![Page 11: Www.le.ac.uk Vectors 1: An Introduction to Vectors Department of Mathematics University of Leicester.](https://reader036.fdocuments.us/reader036/viewer/2022081507/5519b0485503465b578b4574/html5/thumbnails/11.jpg)
Dimensions
• We live in a three dimensional world; can you give a reason why?
Next
What is a Vector? Dimensions Basis
Drawing Vectors
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Basis
A basis is a set of linearly independent vectors which spans a space. This means any point in the space can be written as a combination of scalar multiples of the basis vectors.
Next
What is a Vector? Dimensions Basis
Drawing Vectors
![Page 13: Www.le.ac.uk Vectors 1: An Introduction to Vectors Department of Mathematics University of Leicester.](https://reader036.fdocuments.us/reader036/viewer/2022081507/5519b0485503465b578b4574/html5/thumbnails/13.jpg)
Basis Vectors
Let’s think about the basis vectors of the 2D space.
Any point in this space can be described as the vector .
We can also write this in the following way:
y
x
1
0
0
1yx
y
x
Next
What is a Vector? Dimensions Basis
Drawing Vectors
![Page 14: Www.le.ac.uk Vectors 1: An Introduction to Vectors Department of Mathematics University of Leicester.](https://reader036.fdocuments.us/reader036/viewer/2022081507/5519b0485503465b578b4574/html5/thumbnails/14.jpg)
Basis Vectors
• We say the vectors and
are basis vectors of the 2D space.
• In a similar way, , ,
are basis vectors of the 3D space.
• e1, e2 and e3 are also called i, j and k.
0
11e
1
02e
0
0
1
1e
0
1
0
2e
1
0
0
3e
Next
What is a Vector? Dimensions Basis
Drawing Vectors
![Page 15: Www.le.ac.uk Vectors 1: An Introduction to Vectors Department of Mathematics University of Leicester.](https://reader036.fdocuments.us/reader036/viewer/2022081507/5519b0485503465b578b4574/html5/thumbnails/15.jpg)
Question….
Which of the following is equivalent to ?
kji 42
12
4
kji 42
kji 24
What is a Vector? Dimensions Basis
Drawing Vectors
![Page 16: Www.le.ac.uk Vectors 1: An Introduction to Vectors Department of Mathematics University of Leicester.](https://reader036.fdocuments.us/reader036/viewer/2022081507/5519b0485503465b578b4574/html5/thumbnails/16.jpg)
( )
Vector in two dimensions
x
yab
v
v =
v = i+ ja b
Here we find the position relative to the origin
Click here to see how the vector
is formed.
Next
What is a Vector? Dimensions Basis
Drawing Vectors
![Page 17: Www.le.ac.uk Vectors 1: An Introduction to Vectors Department of Mathematics University of Leicester.](https://reader036.fdocuments.us/reader036/viewer/2022081507/5519b0485503465b578b4574/html5/thumbnails/17.jpg)
x
y bjaib
av
![Page 18: Www.le.ac.uk Vectors 1: An Introduction to Vectors Department of Mathematics University of Leicester.](https://reader036.fdocuments.us/reader036/viewer/2022081507/5519b0485503465b578b4574/html5/thumbnails/18.jpg)
x
y
v
a
b Click here to Repeat
Click here to Go Back
bjaib
av
![Page 19: Www.le.ac.uk Vectors 1: An Introduction to Vectors Department of Mathematics University of Leicester.](https://reader036.fdocuments.us/reader036/viewer/2022081507/5519b0485503465b578b4574/html5/thumbnails/19.jpg)
Try drawing some vectors
x
y
v =
v = i + j
2
-2
4
6
8
-6
-8
-4
0-2-6 -4 2 4 6 8-8
Next
What is a Vector? Dimensions Basis
Drawing Vectors
Draw this
Draw this
Clear
![Page 20: Www.le.ac.uk Vectors 1: An Introduction to Vectors Department of Mathematics University of Leicester.](https://reader036.fdocuments.us/reader036/viewer/2022081507/5519b0485503465b578b4574/html5/thumbnails/20.jpg)
x
z abc
v =
yv
v = i+ j+ k
a b c
Vector in three dimensionsClick here to see how
the vector is formed.
Next
What is a Vector? Dimensions Basis
Drawing Vectors
![Page 21: Www.le.ac.uk Vectors 1: An Introduction to Vectors Department of Mathematics University of Leicester.](https://reader036.fdocuments.us/reader036/viewer/2022081507/5519b0485503465b578b4574/html5/thumbnails/21.jpg)
x
z
y
ckbjai
c
b
a
v
![Page 22: Www.le.ac.uk Vectors 1: An Introduction to Vectors Department of Mathematics University of Leicester.](https://reader036.fdocuments.us/reader036/viewer/2022081507/5519b0485503465b578b4574/html5/thumbnails/22.jpg)
x
z
yv
Click to here repeat
Click here to go back
ckbjai
c
b
a
v
a b
c
![Page 23: Www.le.ac.uk Vectors 1: An Introduction to Vectors Department of Mathematics University of Leicester.](https://reader036.fdocuments.us/reader036/viewer/2022081507/5519b0485503465b578b4574/html5/thumbnails/23.jpg)
kji 5 kji 5kji 5
Which of the following is equivalent to this vector:
x
z
y
v5
1
1
What is a Vector? Dimensions Basis
Drawing Vectors
![Page 24: Www.le.ac.uk Vectors 1: An Introduction to Vectors Department of Mathematics University of Leicester.](https://reader036.fdocuments.us/reader036/viewer/2022081507/5519b0485503465b578b4574/html5/thumbnails/24.jpg)
Conclusion
Vectors are quantities that have a size and a direction.
They can be in any number of dimensions.
The number of basis vectors required is equal to the number of dimensions.
Next
What is a Vector? Dimensions Basis
Drawing Vectors
![Page 25: Www.le.ac.uk Vectors 1: An Introduction to Vectors Department of Mathematics University of Leicester.](https://reader036.fdocuments.us/reader036/viewer/2022081507/5519b0485503465b578b4574/html5/thumbnails/25.jpg)