Vectors• A vector is basically an arrow that represents

the magnitude and direction of a measurement.

• The length of the vector represents its magnitude.

• The direction of the arrow represents the direction in whatever coordinate system is in use.

Notation• A vector quantity is usually represented as

the variable in question with an arrow over it. Sometime it may be written in boldface rather than with an arrow.

• The scalar value or magnitude that relates to a vector (i.e., the length) can be represented as the variable of the vector quantity without the arrow (also not in boldface) or as that variable placed inside an absolute value bracket.

Coordinate systemsCartesian Coordinates

X and Y directions are rectilinear

Cartesian Coordinates

• We can represent a vector in Cartesian coordinates by giving its end point (the tip of the arrow) as an ordered pair (2D) or ordered triple (3D)

(x,y) (x,y,z)(3,4) (3,4,5)

Sketch These!

Now, another method…• We can represent a vector in Cartesian

coordinates by showing the Resultant as the sum of x,y, and z component unit vectors.

• A unit vector is simply a vector of magnitude (length) 1 unit in a given direction. In this case, the x-, y-, and z-directions, respectively.

• Sometimes instead of x,y,z we instead use i,j,k to represent the same thing.

Polar CoordinatesRelate a radius and an angle of incline

(r,)(12 m, 30o)12 m @ 30o

12 m @ 30o N of E12 m @ 60o E of N-12 m @ 30o S of W-12 m @ 210o

These are all the same vector!

PracticeSketch the following vectors.

1) R = -3x + y

2) R = 10x

3) R = 3y

4) R = 2y + 6z

5) R = i + 3j

6) R = 2j – 4i

7) R = 4i – 2j

8) R = x + 2y + 3z

Practice (cont’d) Sketch the following vectors

1) 30 m/s @ 15o N of E

2) 12 m E

3) 9.8 m/s2 down

4) 50 m/s @ 130o

5) 50 m/s @ 50o N of W

6) -13 m @ 10o E of S

7) 22 m/s @ -20o

9) 22 m/s @ 20o

10) r = 6m @ 35o

11) r = 3m @ 270o

12) r = -9m

Pythagorean Theorm222 cba 22 bac

If we let the hypotenuse of a right triangle represent the a vector, the legs of that triangle represent the horizontal and vertical components of that vector.

This allows us to break a vector down to find out its magnitude in the horizontal and vertical directions.

Why do you think this would be important?

Resolution of independent vector components

• We use this technique in physics because as you will learn shortly, vectors of the same variable that are at right angles to each other do not have any effect on each other.

• That is, motion in the horizontal direction does not have any effect on motion in the vertical direction.

Unfortunately...We need trigonometry to do this...

Side a corresponds to angle A

Side b corresponds to angle B

Side c corresponds to angle C

sin = opp/hypcos = adj/hyptan = opp/adj

sin A = a/ccos A = b/ctan A = a/b

sin B = b/ccos B = a/ctan B = b/a

So if you know any two angles, and any two sidesyou can extrapolate the rest of the triangle.

ExampleIf a car travels at 50 m/s at 20o North of East, find the horizontal(east) and vertical (north) components of the velocity.

Rx = Rcoso

Rx = 50cos20o

Ry= Rsino

Ry = 50sin20o

BE CAREFUL!!!

You will usually use sin for your “y” component and cos for your “x”

component, but it always depends on the orientation of the given angle

within the system!!

Write out your trig def’ns if you are not sure!!

Law of Sines

c

C

b

B

a

A sinsinsin

This relationship will allow you to solve ANY trianglelong as you know at least 1 side and 2 angles, or 2 sides and 1 angle. This is very handy for right triangles since you always know at least 1 angle (90o) and have the Pythagorean theorem available.

Adding vectorsWhen adding vectors, place the 1st vector at the origin.

Next place the tail end of one vector to the head end of the other.

Then draw the resultant vector from the origin to the tip of the second vector.

Or, you can add the components and get an exact result.5 m @ 15o N of E + 7 m @ 20o N of E

5 m @ 15o N of E + 7 m @ 20o N of W

5 m @ 15o N of E + 7 m @ 20o S of E

Split each vector into x and y components. Then add the x and y components separately.Now combine the x and y components and find the resultant vector using the Pythagorean theorem to find the magnitudeand trig to find the angle.

More practice:Relative velocities

1) Find the resultant velocity of a boat that crosses a river due east @ 4 m/s while the current runs south @ 1 m/s.

2) What is the displacement of a plane that flies south for 3.0 hours at 500 km/h with a 20 km/h tailwind? A 15 km/h headwind?

3) A cannonball is shot upwards at an angle of 30o above the horizontal with a velocity of 35 m/s. Find the horizontal and vertical components of the velocity. Draw these component vectors.

4) A car drives down a street at 30 m/s. A man is walking in the same direction as the car at 2 m/s as he passes a stationary mailbox. What is the velocity of the car with respect to the man? The car with respect to the mailbox? The man with respect to the car?

4) An evil physics student fires a potato gun forward out of a truck traveling at 25 m/s. If the gun propels the potato at 52 m/s, how fast is the potato traveling

a) When it strikes a stationary parked car?

b) When it strikes a cyclist riding forward at 4 m/s?

c) When it strikes a cyclist riding “backwards” relative to the truck at 4 m/s?

d) Now the potato was fired backwards off of the truck. Do a), b) & c) for this case.

5) Train A heads east at 175 m/s. Train B heads west at 150 m/s. What is the velocity of train A with respect to train B?

6) The trains in problem 5 are now both traveling north.

What is the velocity of train B with respect to train A?

What is the velocity of train A with respect to train B?