VectorsAn Introduction

There are two kinds of quantities…•Vectors are quantities that

have both magnitude and direction (e.g., displacement, velocity, acceleration).

•Scalars are quantities that have magnitude only (e.g., position, speed, time, mass).

Notating vectors

RThis is how you draw a vector.

•Vector: R R→

headtail

Notating scalars•Scalar: RThere is no standard way to draw a scalar!

Direction of VectorsA

xθ

Bxθ

Vector angle ranges

x

y

θ

I0 < θ < 90o

II90o < θ < 180o

θ

III180o < θ < 270o

θIV

270o < θ < 360o

θ

Magnitude of Vectors• The best way to describe the magnitude of a vector is to measure the length of the vector.

• The length of the vector is proportional to the magnitude of the quantity it represents.

Magnitude of VectorsA

If vector A represents a displacement of three miles to the north… B

Then vector B, which is twice as long, would represent a displacement of six miles to the north!

Equal VectorsEqual vectors have the same length and direction, and represent the same quantity (such as force or velocity).

Inverse VectorsInverse vectors have the same length, but opposite direction.

A

-A

Vectors: x-component

Ax = A cos θ

A

θ xAx

Vectors: y-component

Ay = A sin θ

A

θ x

Ay

Vectors: angle

θ = tan-1 (Ry/Rx)

x

y

Rx

Ry

θ

Vectors: magnitude

R = √ (Rx2 + Ry

2)

x

y

Rx

Ry

R

You’ll need:Graph paperPencilsRulerProtractor

Graphical Addition of Vectors

A

B

R A + B = R

Graphical Addition of Vectors

R is called the resultant vector!

The Resultant and the Equilibrant

The sum of two or more vectors is called the resultant vector.

The resultant vector can replace the vectors from which it is derived.

The resultant is completely canceled out by adding it to its inverse, which is called the equilibrant.

A

B

R A + B = R

Graphical Addition of Vectors

E is called the equilibrant vector!

E

Component Addition of Vectors

1) Resolve each vector into its x- and y-components.Ax = Acosθ Ay = AsinθBx = Bcosθ By = BsinθCx = Ccosθ Cy = Csinθ

etc. Add the x-components (Ax, Bx, etc.)

together to get Rx and the y-components (Ay, By, etc.) to get Ry.

Component Addition of Vectors

3) Calculate the magnitude of the resultant with the Pythagorean Theorem R = √(Rx

2 + Ry2)

4) Determine the angle with the equation θ = tan-1 Ry/Rx.

Relative Motion

Vs

Vt = Vs + Vw Vw

S = swimmerW = water

Relative Motion

Vs

Vt = Vs + Vw Vw

Relative Motion

Vs

Vt = Vs + Vw Vw