PreCalculus Notes: Vectors



PreCalculus Notes Vectors -1: Vectors

Definitions

Scalar: A quantity/measurement that has magnitude (size) but no direction.

Ex: Distance (length), speed, time, mass, temperature, population, price, GNP, IQ, etc.

Vector: A quantity/measurement that has both magnitude and direction.

Ex: Displacement: Displacement is change in position; a distance and a direction.

Velocity: Velocity is rate of change of position; speed with direction.

Force

In physics, all vectors have units; e.g.: m, m/s, kg-m/s2 (aka newtons), etc.

Notation

Two common notations:

1. Often in printed material (textbooks, etc), vectors are indicated by bold letters: x, r, v, F, etc.

2. In writing (and sometimes in print), vectors are indicated by arrows over the letter: [pic], [pic], [pic], [pic]

Geometric Representation

Vectors are represented geometrically as “directed line segments.” (Note: not rays.)

The length of the segment represents its magnitude (size).

The slope of the segment and the arrow represent its direction.

The end with the arrow is called the tip or the head of the vector.

The end without the arrow is called the tail.

Note: Vectors have size and direction but no fixed position. In the diagram,

the “two” vectors labeled [pic]are two “representatives” of the same vector.

Analytic Representation; Components

To represent a vector analytically (numerically/algebraically), we need a coordinate system.

There are two common ways to represent a vector [pic]:

1. Length and direction angle

v (no arrow) or ||[pic]|| indicates the magnitude (length) of the vector

( indicates the direction, counterclockwise from the positive x-axis

2. Components: [pic] = (vx, vy( = (x2 ( x1, y2 ( y1(

Ex: Suppose [pic] has its tail at (5, 3) and its tip at (1, 6).

a. Find the components of [pic].

Notes:

1. Components of a vector are not the same as an

ordered pair. However, if the “initial point” (x1, y1) is

the origin, then the components (vx, vy( of the vector are

the same as the coordinates of the “terminal point” (x2, y2).

2. A vector with its “tail” at the origin is in “standard position.”

3. In physics, a vector with its tail at the origin is often called

a “position vector,” typically denoted [pic].

b. Find the magnitude [pic].

c. Find the direction angle of [pic].

Note: We need to be careful to choose ( in the correct quadrant.

Ex: A vector has magnitude 13 and direction angle (22.62(.

Find its components.

Summary: Length, Direction Angle and Components

If we know v and (, then

If we know vx and vy, then

Ex: Bobby Boy Scout walks 4 km at a heading of 125(. Represent his displacement vector in component form.

Ex: Betsy Bluebird is flying at 8 m/s on a heading of 206(. Represent her velocity in component form.

Ex: Bertha the Bear walked from her den at ((1, 2) to the honey tree at (2, (3).

a. Represent her displacement vector in component form.

b. Find the magnitude and direction of Bertha’s displacement.

Note: Displacement means “change in position,” the difference between where Bertha started and where she ended up. It is a vector, with both a size and a direction. Distance is a scalar that tells how far Bertha actually walked to get from where she started to where she ended up. Distance is not necessarily the same as the magnitude of displacement. This will only happen if Bertha walks in a straight line.

PreCalculus Notes Vectors -2: Addition and Subtraction of Vectors

Geometric Representation

Ex: On the diagram, represent [pic], [pic], [pic] and [pic]

Component Form

Ex: (Same as above.) In component form, [pic] and [pic] .

[pic]

[pic]

Ex: Bertha the Bear ambled 4 km on a heading 60( and then turned and moseyed 6 km on a heading

350(. Where did Bertha end up?

Ex: Bobby Boy Scout paddles his canoe due east across the Big River at a steady 3 km/hr relative to the water. The current in the river moves at 2 km/hr on a heading of 160(. How fast and in what direction is Bobby actually traveling?

Ex: Betsy Bluebird wants to travel to her nest, which is 90 meters away on a heading of 328(. After flying on the correct heading for 10 seconds at 8 m/s relative to the air, Betsy finds to her dismay that she has actually flown 50 meters on a heading of 310(.

a. What is the speed and direction of the wind?

b. How far and in what direction is Betsy’s nest from her current location?

Ex: A boat leaves port wishing to travel to a second port 800 nm away on a heading of 340°. Some time later the captain finds that due to navigation error, the ship has actually traveled 500 miles on a heading of 335°. What direction and how far should the ship sail to reach its original destination?

PreCalculus Notes Vectors - 3: Forces

Newton’s Second Law: [pic] where [pic] = Net force; m = mass; [pic]= acceleration

1. If [pic] = 0, then

- object is either “at rest” or maintaining constant velocity (same speed, same direction)

- the vector sum of all the forces on the object must be 0

2. If [pic] ( 0, then

- object is accelerating (changing speed, direction or both)

- there must be a net positive force in the direction of acceleration.

Ex: A kid is on a swing supported by two ropes each 10 feet long. The combined weight of kid and swing is 180 newtons. Big brother pulls the swing back a horizontal distance of 6 feet. What horizontal force must he apply to keep the swing (and kid) in that position? What is the tension in the rope(s) holding the swing?

Ex: A boat and trailer together weigh 1800 lbs. They are on a ramp that slopes up 20( from the horizontal. What force must be exerted along the ramp to keep the boat and trailer moving at a constant speed?

Ex: A 300 N weight is suspended from the ceiling by two ropes. One rope makes an angle of 30( with the vertical; the other rope makes an angle of 40( with the vertical. Find the tension in each rope.

PreCalculus Notes Vectors - 4: Forces, Again

Ex: Three hyenas, Snarly, Carly and Farly, are fighting over the carcass of a small goat. Snarly is pulling on the goat with a force of 50 newtons (N). Carly is pulling with a force of 30 N at an angle of 110( to the direction Snarly is pulling. Farley is pulling at 40 N in a direction (150( from Snarly’s direction. What is the net force on the goat?

Ex: Liz is skiing down a hill inclined at 30( to the horizontal. Liz weighs 330 N and the kinetic coefficient of friction between her skis and the snow is 0.15. What is the net downhill force on Liz?

Ex: Anthony is dragging a box weighing 450 N up a ramp inclined at 20( to the horizontal. Anthony is dragging the box with a rope that makes a 30( with the ramp. The coefficient of kinetic friction between the box and the ramp is 0.7. Find the tension Anthony must maintain in the rope to keep the box moving up the ramp at a constant speed.

PreCalculus Notes Vectors - 6: Dot Products

Definition: If u = (u1, u2( and v = (v1, v2(, then the dot product (a.k.a inner product, a.k.a. scalar product) of

u and v is

u ( v = u1v1 + u2v2 Note: This is a scalar.

Ex: u = [pic] and v = [pic],

a. Find u ( v

b. Find u ( u

Properties of Dot Product

1. u ( v =

2. u ( v = v ( u

3. u ( (v + w) = u ( v + u ( w

4. u ( u =

Angle between two vectors

Ex: Find the angle between u = [pic] and v = [pic].

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v

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6

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x

v

vy

vx

u

[pic]

(

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10

v

vx

y

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[pic]

(x2, y2)

y

x

vy

[pic]

[pic]

To add vectors graphically: move the second vector so that it begins where the first one ends. The sum, called the resultant, goes from the beginning of the first vector to the end of the second. To subtract, simply add the opposite (the opposite of a vector has the same length but the opposite direction). (Note:

u + v and v + u are the same; addition is commutative.)

[pic]

(x1, y1)

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