Vectors

Chapter 3, Sections 1 and 2

Vectors and Scalars

• Measured quantities can be of two types• Scalar quantities: only require

magnitude (and proper unit) for description. Examples: distance, speed, mass, temperature, time

• Vector quantities: require magnitude (with unit) and direction for complete description. Examples: displacement, velocity, acceleration, force, momentum

Vector Addition

• When 2 or more vectors act on an object, the total effect is the vector sum

• Special math operations must be used with vectors

• Vector sum called the resultant• Sum can be found using graphic

methods (drawing to scale) or mathematical methods

Adding Vectors Graphically• Choose a suitable scale for the

drawing• Use a ruler to draw scaled magnitude

and a protractor for the direction• Vectors can be moved in a diagram

as long as their length and direction are not changed

• Vectors can be added in any order without changing the result

Adding Vectors Graphically

• Use head-to-tail method for series of sequential vectors where each successive vector begins where the preceding vector ended

• Also works for two vectors acting simultaneously at the same point, although drawing doesn’t match the physical situation

Adding Vectors Graphically

• Resultant is a vector drawn from point of origin to tip of last vector

• Magnitude of resultant can be found by measuring and converting the measurement using the scale of the drawing

• Direction is found by measuring angle with a protractor

Adding Vectors Graphically

• Graphical method gives approximate values depending on drawing accuracy

One dimensional graphical addition of vectors

Subtracting Vectors

• To subtract a vector, add a negative vector, one having same magnitude but opposite direction

ab

a +b

a - b

a

a

b

-b

Parallelogram Method

• Vectors are drawn from a common origin

• Complete parallelogram by drawing opposite sides parallel to vectors

• Resultant is the diagonal of the parallelogram

Pros and Cons for Graphical Methods

• For simultaneous vectors like forces, parallelogram method gives a better picture of actual situation

• More difficult to draw accurately• Better for sketches, not for

measured drawings• Head-to-tail method better for

measuring

Parallelogram Method

• a + b = r

a

b

r

Adding Vectors Mathematically

• Exact values for vector sums using trig functions (tan mostly) and Pythagorean theorem

• Set up vectors on x-y coordinate system• If vectors act at right angles,

Pythagorean theorem gives resultant magnitude

• Direction can be found with tan-1 function

Resolving Vectors Into Components• A vector acting at an angle to the

coordinate axes can be resolved into x and y components that would add together to equal the original vector

• The x-component = original magnitude times the cos of the angle measured from the x-axis

• The y-component = original magnitude times the sin of the angle measured from the x-axis

Vector Components

vx= (50 m/s)(cos 60o)

vy= (50 m/s)(sin 60o)

Adding Non-perpendicular Vectors

• Resolve each vector into x and y components

• Add the x-components together and add the y-components together

• Use Pythagorean theorem and tan-

1 function to find magnitude and direction of resultant

Adding Non-perpendicular Vectors

x

y

x

y

Adding Non-perpendicular Vectors

Rx = 11.3 + 12.5 = 23.8

Ry= 4.1 + 21.7 = 25.8

2 223.8 25.8 35R

@ 1 25.8tan 47

23.8

Alternate Method for Adding Non-perpendicular Vectors

• Consider a vector triangle with angles A, B, and C with opposites sides labeled a, b, and c

b

a

c

BC

A

Alternate Method for Adding Non-perpendicular Vectors

• Cosine law can be used to find magnitude of vector c if magnitudes and directions of a and b are known

• Angle between a and b can be found using simple geometry

• Sine law can be used to find direction of vector c

Cosine Law

• Useful if two sides (a and b) and the angle between them (C) are known:

• Similar to Pythagorean Theorem with a correction factor for lack of right angle

2 2 2 cosc a b ab C

Sine Law

• Useful when one angle and its opposite side are known along with one other side or angle

sin sin sin

a b c

A B C