MATH Vector combinations

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MATH 160

1.2 Vector combinations

David Bryant

University of Otago

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Notices

Online (recorded) lectures all this week. I’ll post them as soon as they are completed. Next lockdown I’ll ask for live

streaming but it was impossible this time around.

Printed course notes will be available at the print shop in the main library, but not yet.

Quizzes and tutorials start next week.

Sorry for the abrupt ending to the video yesterday. Forgot to tidy that up before posting.

Check the discussion board on blackboard for more info, as well as the course webpage at maths, and the course outline on the resources page.

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Summary of lecture 1.1

Lots of blah blah about applications (too much?)

Vectors in R2: the notation R2 says we are dealing with pairs of real numbers. We print these as column arrays.

Vectors in R3 and Rn: vectors with three, or n, real entries. Transpose

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Representing vectors

There are two standard geometric ways to represent vectors: Representing vectors as points;

Representing vectors as arrows.

Both approaches have their advantages and disadvantages. Different mathematicians / scientists emphasise different representations. Chances are, you were introduced to vectors as arrows.

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3Blue1Brown

Check out 3b1b channel on linear algebra.

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Point representation

Every vector corresponds to a unique point, with coordinates given by the entries in the vector.

Example 1.1

Draw a point representation for u = 2 2 , v = −2 −1 , w = 3 −1

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Arrow representation

The other main way of representing vectors geometrically is using arrows. To represent a vector like u =

x y

we Choose an arbitrary location for the tail.

Draw an arrow to the point that is x units across and y units up from the tail.

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Example 1.2

Draw an arrow representation for u = 2 2 , v = −2 −1 , w = 3 −1

where all tails are at the origin. Example 1.3

Draw an arrow representation for u = 2 2 , v = −2 −1 , w = 3 −1

where all tails are at arbitrary positions.

When a vector is represented by an arrow, it is the length and direction of this arrow that is important, not the position.

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Vector clones

Different arrows can represent equal vectors!

u

v w

u

v

w _u

Two arrows that are translations of each other (same di-rection, length, but different positions) represent the same vector.

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Notation

Let P and Q be two points in R2. You’ll sometimes see the notation

−→ PQ

used to denote the vector corresponding to an arrow with tail at P and head at Q.

P Q

! P Q

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Adding vectors

Given two vectors u and v in Rn, their sum is the vector obtained by adding the corresponding entries of u and v.

1 −2 + 2 5 = 1 + 2 −2 + 5 = 3 3 . The sum of two vectors with different sizes is undefined. Example 1.4

Compute the following sums   1 3 −1   +   0 −1 −1   0, 0, 0, 0T +1, 2, 2, 1T   1 2 3   + 4 5 .

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Scalar multiplication

The scalar multiple of a vector u by a number c is the vector obtained by multiplying each entry in u by c. For instance,

if u = 3 −1 and c = 5, then cu = 5 3 −1 = 15 −5 . The number c in cu is called a scalar (because it scales).

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Combining scalar multiplication and addition

Suppose u = 1 −2 and v = 2 −5 . Compute 4u + (−3)v: 4u = 4 −8 (_{−3)v =} −6 15 4u + (_{−3)v =} 4 −8 + −6 15 = −2 7 . Example 1.5 Let a =  30 1  , b =   23 −1  , c =  −23 0  . Compute

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Math 160 Exam, 2018

15 MATH 160 1.Let x = [1 mark] 2 414 3 3 5 and y = 2 411 2 3

5. Which of the following equals 3x 2y?

A. 2 405 1 3 5 B. 2 423 5 3 5 C. 2 4105 13 3 5 D. 2 4105 13 3 5 E. 2 453 0 3 5 Solution: D

2.This is an arrow representation of three vectors u, v and w. [1 mark] u

v

w

Which of the following vectors equals zero? A. u + v + w B. u w + v C. u w v D. v + w u E. u + w v Solution: B

3.Find the value of c such that the vectors⇥4c 1 1⇤Tand⇥2 c 9⇤Tare [1 mark] orthogonal.

A. c = 1 B. c = 1 C. c = 0 D. c = 9

E. None of the above Solution: B

Stop the video now and complete this question. (I’ll record solutions to these after the lecture).

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Effect of scaling: point representation

Example 1.6 Let u = 2 2 , v = −2 −1 , w = 3 −1

and c = 1.5. Plot cu, cv and cw using the point representation.

The effect of this scalar multiplication is to move a point closer or further away from the origin along the line through the origin and that point.

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Effect of scaling by a negative: points

Example 1.7 Let u = 2 2 , v = −2 −1 , w = 3 −1

and c =_{−1. Plot cu, cv and cw using the point representation.}

The effect of scaling by_{−1 is to reflect the point through the} origin.

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Effect of addition: points

Example 1.8 Let x = 2.5 −0.5 y = 1 2 . Plot x + y.

If x and y are represented as points in the plane, then x + y corresponds to the fourth vertex of the parallelogram whose other vertices are x, 0, y.

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We have seen how scalar multiplication and addition works in the point representation. Now the arrow representation.

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Scaling vectors: arrows

Suppose u = 2 2 , v = −2 −1 , w = 3 −1 and suppose that c = 1.5. Then

cu = 3 3 , v = −3 −1.5 , w = 4.5 −1.5 u v w cu cw cv

The effect of the scalar multiplication has been to stretch all of the arrows.

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Scaling vectors by a negative: arrows

u = 2 2 , v = −2 −1 , w = 3 −1

and suppose that c =_−1. cu = −2 −2 , cv = 2 1 , cw = −3 1 u v w u u v w u

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Scalar multiplication: arrows

Effect of scaling by c:

c > 1 Stretches arrows longer

c = 1 No change

0< c < 1 Shrinks arrows shorter

c = 0 Shrinks arrow to a dot

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Addition: arrows

Example 1.9

Compute and plot an arrow representation of x + y when x = 2.5 −0.5 y = 1 2 .

To find x + y move the tail of y to the head of x. Then x + y is the arrow from the tail of x to the head of y.

y 0

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MATH 160 Exam, 2018

A. 2 405 1 3 5 B. 2 423 5 3 5 C. 2 4105 13 3 5 D. 2 4105 13 3 5 E. 2 453 0 3 5 Solution: D

v

w

A. c = 1 B. c = 1 C. c = 0 D. c = 9

Stop the video now and complete this question. (I’ll record solutions to these after the lecture).

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Online octave

Octave is a rip-off of the mathematical software Matlab, which is the industry standard for linear algebra.

You don’t need to learn Octave for this course, but I’ll use it occasionally in lectures.

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Vectors in octave

Variable names start with a letter and are made up of letters and numbers. e.g. u,v,w

To specify a vector, use square brackets, and separate lines with ; u = [3;-1];

v = [0.2;0.3]; w = [4;-7.2];

You can then add and scale vectors: 3*u - 2*v

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Math 160 Exam, 2018 - solution

A. 2 405 1 3 5 B. 2 423 5 3 5 C. 2 4105 13 3 5 D. 2 4105 13 3 5 E. 2 453 0 3 5 Solution: D

v

w

A. c = 1 B. c = 1 C. c = 0 D. c = 9

E. None of the above Solution: B We have 3x =   3 12 −9   and (−2)y =   2 −2 −4   so adding these, we get

3x− 2y =   5 10 −13   or option D. David Bryant

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MATH 160 Exam, 2018 - solution

A. 2 405 1 3 5 B. 2 423 5 3 5 C. 2 4105 13 3 5 D. 2 4105 13 3 5 E. 2 453 0 3 5 Solution: D

v

w

A. c = 1 B. c = 1 C. c = 0 D. c = 9

You need two rules:

To add to vectors, put the tail of one on the head of the other. To subtract a vector, first flip it (swap head and tail) and then add it.

On the next page I give the five vector combinations. The zero one will start and end in the same place (option B)

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MATH 160 Exam, 2018 - solution (2)

u v w u v w u v w u v w u v w A B C D E -u v w