6.1 – Vectors in the Plane

What are Vectors?

Vectors are a quantity that have both magnitude (length) and direction, usually represented with an arrow:

• This includes force, velocity, and acceleration

• Component Form:v = <-2,3>

Naming Vectors

A vector can also be written as the letters

of its head and tail with an arrow above:

A – initial pointB – terminal point 

Scalars

A quantity with magnitude alone, but no directions, is not

a vector, it’s called a scalar

For example, the quantity “60 miles per hours” is a

regular number, or scalar. The quantity “60 miles per hour

to the northwest” is a vector, because it has both size and

direction

Components

To do computations with vectors, we place them in the plane and find their components.

v

(2,2)

(5,6)

Components

The initial point is the tail, the head is the terminal point. The components are obtained by subtracting coordinates of the initial point from those of the terminal point.

v

(2,2)

(5,6)

Components

The first component of v is 5 -2 = 3. The second is 6 -2 = 4. We write v = <3,4>

v

(2,2)

(5,6)

Magnitude of a Vector

The magnitude (or length) of a vector is shown by two vertical bars on either side of the vector: |a|

OR it can be written with double vertical bars: ||a||

1 1 2 2

2 2

2 1 2 1

2 2

If is represented by the arrow from , to , , then

.

If , , then .

x y x y

v x x y y

a b a b

v

v v

Magnitude of a Vector

Find the magnitude of the vector:

V = <-2,3>

Finding Magnitude of a Vector

Find the magnitude of represented by , where (3, 4) and

(5,2).

PQ P

Q

v

2 2

2 1 2 1

2 2

5 3 2 ( 4)

2 10

x x y y

v

Showing Vectors are Equal

Let u be the vector represented by the directed line segment from R to S, and v the vector represented by the directed line segment from O to P. Prove that u =v.

Addition

To add vectors, simply add their components.

For example, if v = <3,4> and w = <-2,5>,

then v + w = <1,9>.

Multiples of Vectors

Given a real number c, we can multiply a vector by c by multiplying its magnitude by c:

v2v -2v

Notice that multiplying a vector by anegative real number reverses the direction.

Scalar Multiplication

To multiply a vector by a real number, simply multiply each component by that number.

If v = <3,4> and w = <-2,5>, then:

-2v =

4v – 2w =

Vector Operations Example

Let 2, 1 and 5,3 . Find 3 . u v u v

Vector Operations Example

Let 2, 1 and 5,3 . Find 3 . u v u v

3 3 2 , 3 1 = 6, 3

3 = 6, 3 5,3 6 5, 3 3 11,0

u

u v

Unit Vectors

A unit vector is a vector with magnitude (length) of 1.

Given a vector v, we can form a unit vector by multiplying the vector by 1/||v||.

Or you can think of this as v/||v|| (The vector divided by its magnitude)

Finding a Unit Vector

Find a unit vector in the direction of 2, 3 . v

Finding a Unit Vector

Find a unit vector in the direction of 2, 3 . v

222, 3 2 3 13, so

1 2 32, 3 ,

13 13 13

| |

| |

v

v

v

Standard Unit Vectors

A vector such as <3,4> can be written as 3<1,0> + 4<0,1>.

For this reason, these vectors are given special names: i = <1,0> and j = <0,1>.

A vector in component form v = <a,b> can be written ai + bj.

For example, rewrite the vector <-3, 2>

Direction Angles

The precise way to specify the direction of a vector is to state its direction angle (not its slope).

v

Direction Angles

If has direction angle , the components of can be computed

using the formula = | | cos , | | sin .

From the formula above, it follows that the unit vector in the

direction of is cos ,sin .| |

v v

v v v

vv u

v

Finding the components of a Vector

Find the components of the vector with direction angle 120 and

magnitude 8.

v

Finding the components of a Vector

Find the components of the vector with direction angle 120 and

magnitude 8.

v

, 8cos120 ,8sin120

1 3 8 ,8

2 2

4,4 3

So 4 and 4 3.

a b

a b

v

Examples

Find the component form of v, with magnitude 15 and a direction angle of 40 degrees.

Find the component form of vector v with magnitude 6 and direction angle of 115 degrees.

Examples

Find the component form of v, with magnitude 15 and a direction angle of 40 degrees.

<15 cos 40, 15 sin 40> = <11.491, 9.642>

Find the component form of vector v with magnitude 6 and direction angle of 115 degrees.

<6 cos 115, 6 sin 115> = <-2.536, 5.438>

Finding the Direction Angle of a Vector

Find the magnitude and direction angle of 2,3 .u

Finding the Direction Angle of a Vector

2 2|| || 2 3 13

Let be the direction angle of , then

2,3 13 cos , 13sin

2 13 cos

56.3

u

u

u

Find the magnitude and direction angle of 2,3 .u

Finding the direction angle

Find the direction angle for the vector <8, -4>

v

Velocity and Speed

The velocity of a moving object is a vector because velocity has both magnitude and direction.

The magnitude of velocity is speed.

Word Problem

An airplane is flying on a compass heading (bearing) of 170 degrees at 460 mph. A wind is blowing with a bearing of 200 degrees at 80 mph.

a)Find the component form of the velocity of the airplane

b)Find the actual ground speed and direction of the plane.