Vectors

Vectors and Scalars• Vector: Quantity which requires both

magnitude (size) and direction to be completely specified– 2 m, west; 50 mi/h, 220o

– Displacement; Velocity

• Scalar: Quantity which is specified completely by magnitude (size)– 2 m; 50 mi/h– Distance; Speed

Vector Representation

• Print notation: A– Sometimes a vector is

indicated by printing the letter representing the vector in bold face

Mathematical Reference System

x

y

0o

90o

180o

270o

Angle is measured counterclockwise wrt positive x-axis

Equal and Negative Vectors

Vector Addition

A + B = C (head to tail method)

B + A = C (head to tail method)

A + B = C (parallelogram method)

Addition of Collinear Vectors

Adding Three Vectors

Vector Addition Applets

• Visual Head to Tail Addition• Vector Addition Calculator

Subtracting Vectors

Vector Components

Horizontal Component

Ax= A cos

Vertical Component

Ay= A sin

Signs of Components

Components ACT• For the following, make a sketch and

then resolve the vector into x and y components. 60 ,120oA m 40 ,225oB m

Ay

Ax

Ay = (60 m) sin(120) = 52 m

Ax = (60 m) cos(120) = -30 m

Bx

By

Bx = (40 m) cos(225) = -28.3 mBy = (40 m) sin(225) = -28.3

m

(x,y) to (R,)• Sketch the x and y

components in the proper direction emanating from the origin of the coordinate system.

• Use the Pythagorean theorem to compute the magnitude.

• Use the absolute values of the components to compute angle -- the acute angle the resultant makes with the x-axis

• Calculate based on the quadrant*

2 2x yD D D

1tany

x

D

D

360o

*Calculating θ• When calculating the angle, • 1) Use the absolute values of the

components to calculate • 2) Compute C using inverse tangent • 3) Compute from based on the

quadrant.• Quadrant I: = • Quadrant II: = 180o - ; • Quadrant III: = 180o + • Quadrant IV: = 360o -

(x,y) to (R,) ACT

• Express the vector in (R,) notation (magnitude and direction)

A = (12 cm, -16 cm)

A = (20 cm, 307o)

Vector Addition by Components

• Resolve the vectors into x and y components.

• Add the x-components together.

• Add the y-components together.

• Use the method shown previously to convert the resultant from (x,y) notation to (R,) notation

Practice Problem

Given A = (20 m, 40o) and B = (30 m, 100o), find the vector sum A + B.

A = (15.32 m, 12.86 m)B = (-5.21 m, 29.54 m)

A + B = (10.11 m, 42.40 m)

A + B = (43.6 m, 76.6o)

Unit Vectors: Notation

• Vector A can be expressed in several ways

• Magnitude & Direction (A,)• Rectangular Components

(Ax , AY)

yx

ˆ ˆx yA x A yx x

y

y