Vectors

Objectives: • Find the magnitude of the vector• Find component form• Add and subtract vectors

Find the magnitude of a vector

• A vector has both direction and magnitude.

• A vector begins with an initial point, usually P, to a terminal point, usually Q.

• A little reminder for terminal point: to terminate is to end, so the terminal point ends the line.

• Graph it as you would a ray.

• For example at right, the initial point is P(0, 0), and the terminal point is Q(-6, 3).

4

2

-2

-4

-6

-5 5 10

Q(-6, 3)

P(0, 0)

Write the component form

• Write the following Component Form =‹x2 – x1, y2 – y1› or

<u,v><-6 – 0, 3 – 0><-6, 3> is the component form.Next, use the distance formula to find the

magnitude.

|PQ| = √(-6 – 0)2 + (3 – 0)2 = √62 + 32

= √36 + 9 = √45 or 3√5 ≈ 6.7

4

2

-2

-4

-6

-5 5 10

Q(-6, 3)

P(0, 0)

Graph Vector

• Initial point is P(0, 2). Terminal point is Q(5, 4).

• Graph the ray starting at P, and go through Q. See picture. Then start looking for component form and magnitude.

8

6

4

2

-2

5 10 15

Q(5, 4)

P (0, 2)

Write the component form

• Write the following Component Form =‹x2 – x1, y2 – y1› or

<u,v><5 – 0, 4 – 2><5, 2> is the component form.Next use the distance formula to find the

magnitude.

|PQ| = √(5 – 0)2 + (4 – 2)2 = √52 + 22

= √25 + 4 = √29 ≈ 5.4

8

6

4

2

-2

5 10 15

Q(5, 4)

P (0, 2)

Graph Vector

• Initial point is P(3, 4). Terminal point is Q(-2, -1).

• Graph the ray starting at P and going through Q. See picture. Then you can start looking for component form and magnitude.

4

2

-2

-4

-6

5 10

P(3, 4)

Q(-2, -1)

Write the component form

• Write the following Component Form =‹x2 – x1, y2 – y1› or

<u,v><-2 – 3, -1 – 4><-5, -5> is the component form.Next use the distance formula to find the

magnitude.

|PQ| = √-2 – 3)2 + (-1– 4)2 = √(-5)2 + (-5)2

= √25 + 25 = √50 or 5√2 ≈ 7.1

4

2

-2

-4

-6

5 10

P(3, 4)

Q(-2, -1)

Adding Vectors• Two vectors can be added to form a

new vector. • To add u and v on a graph, place

the initial point of v on the terminal point of u.

• ♫ Take note! The sum of the vectors is a new vector from the initial point of u to the terminal point of v.

• You can also add vectors algebraically.

• You may put the initial point of u on the terminal point of v. This has lead to the name “the parallelogram rule”.

What does this mean?

• Adding vectors:

Sum of two vectors

The sum of u = <a1,b1> and v = <a2, b2> is

u + v = <a1 + a2, b1 + b2>• In other words: add your x’s to get the

coordinate of the first number in component form, and add your y’s to get the coordinate of the second number in component form.

• Let u = <3, 5> and v = <-6, 1>• To find the sum vector u + v, add the x’s and

add the y’s of u and v.

u + v = <3 + (-6), 5 + (-1)>

= <-3, 4>

There are 6 of these on the test Wednesday!!!

Example:

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