Vectors

ObjectiveStudents will be able to use basic

vector operations to solve problems.

A vector is a mathematical object that has both magnitude (size) and direction.• A vector is shown as a directed line segment

with initial and terminal points.

Component Form• The component form of a vector is much like

an (x,y) point. It is the horizontal change and vertical change from the initial to the terminal point. ( x,y ) is replaced by

yx,

If (a,b) is point (8,4)Then the component formOf the vector is 4,8

Be careful when the initial point is not at the origin, it is like Counting for slope!

Using matrices with vectors

• A vector with component form <2, -4> can be written as the matrix v =

42

Rotating a Vector

• To rotate a vector write it in matrix form• Multiply by the appropriate rotation matrix.• EX. Rotate the given vector 90 degrees.

The resulting vector is <2,3>

23

32

23

0110

• Try rotating the vector v=• By 270 ◦

53

3

553

0110

The component form is <5,3>

Adding and subtracting vectors

• To add vectors add each corresponding component

• Ex. <-2,3> + <5,-2> = <3, 1>

• To subtract vectors subtract each corresponding component

• Ex. <-2, 7> - < 5, 9> = <-7, -2>

Finding the magnitude of a vector

v

The length (magnitude) of a vector v is written |v|. Length is always a non-negative real number.

Use the distance formula to Find the magnitude of a vector.

2.752

1636

)37()28( 22

Or Pythagorean Theorem

2.752

52

462

222

cc

c

c

Scalar Multiplication with vectors

• Scalar multiplication of a vector by a positive number other than 1 changes the magnitude of the vector.

• Scalar multiplication by a negative number other than -1 changes the magnitude and reverses the direction of the vector.

• For v = < 1, -2 > and w = < 2, 3 > what are the graphs of the following vectors?

• 3v -2w

Finding magnitude and direction.

• Use trigonometry to find the unknown angle.

6.24605

2211 22

A=

4.631122tan 1

Finding Dot Products

• If and• The dot product v◦w is

• If v◦w = 0 the two vectors are normal or perpendicular to each other.

• Ex. Are the following vectors normal? No

21,www 21,vvv

2211 wvwv

3,7,5,2 ut1)3)(5()7)(2( ut

• Try these:• Are these vectors normal?

5,2,4,10 wv 18,9,6,2 wv

Translations and Vectors

Translations and vectors: The translation at the left shows a vector translating the top triangle 4 units to the right and 9 units downward. The notation for such vector movement may be written as:

A fishing boat leaves its home port and travels 150 miles directly east. It then changes course and travels 40 miles due north. How long will the direct return trip take if the boat averages 23 mph.

6.7 hrs

Vectors!

• Bev drove to her friends house 6 blocks north and 5 blocks east. From there, she went to the school gym for volleyball practice 6 blocks north and 8 blocks west.

• A.) If Bev’s house is at the origin, sketch the vectors of her route.

• B.) Describe where the school is in relation to the origin using vector notation in component form.

• C.) Find the magnitude of the sum of the two vectors. Label the sum vector on the graph.

<-3, 12 > 12.68