PHY 1151 Principles of Physics I

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Chapter 3

Vectors in Physics

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Outline Vectors and Scalar Quantities Graphical Representation of

Vectors Some properties of Vectors

Equality of Two Vectors Adding Vectors (Graphically) Subtracting Vectors (Graphically) Multiplying a Vector by a Scalar

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Scalar Quantities Scalar quantity: A scalar quantity

is completely specified by a single value with appropriate units and has no direction. Examples: temperature, volume,

mass, time intervals, speed

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Vector Quantities Vector quantity: A vector quantity is

completely specified by a number with appropriate units (magnitude) plus a direction.

A vector quantity has both magnitude and direction. Examples: position, displacement, velocity,

acceleration, force We use boldface letter, such as A, to

represent a vector quantity. The magnitude of the vector A is written either as A or |A|.

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Graphical Representation of Vectors

Vectors can be represented graphically using arrows The direction of the

arrowhead represents the direction of the vector.

The length of the arrow represents the magnitude of the vector.

Scale: 1 cm 1N

F

Example: a force vector of magnitude = 3 N and directed to the right can be represented as follows

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Examples of Vectors

x

y

0

r

Position Vector r

x

y

0Displacement Vector D as a particle moves from A to B

D

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Equality of Two Vectors Equality of two

vectors: Two vectors A and B are

defined to be equal if they have the same magnitude and point in the same direction.

A = B only if A = B and A and B point in the same direction.

All the vectors in this figure are equal even though they have different starting points.

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Adding Vectors

Graphical representation of vector addition The resultant vector R = A +

B is the vector drawn from the tail of A to the tip of B

The resultant vector R = A + B + C + D is the vector drawn from the tail of the first vector to the tip of the last vector

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Example: Adding Vectors If you walked 3.0 m toward

the east and then 4.0 m toward the north. Find the magnitude and direction of your total displacement.

Two ways to solve the problem: Graphical method: Using

graph paper and a protractor.

Algebra method: Calculation using algebra

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Subtracting Vectors Negative of a vector:

The vectors A and –A have the same magnitude but point in opposite directions. Thus, A + (-A) = 0.

Subtracting vectors: We define operation A – B as

vector – B added to vector A: A – B = A + (- B)

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Multiplying a Vector by a Scalar

If vector A is multiplied by a positive scalar quantity m, then the product mA is a vector, which has the same direction as A and magnitude mA.

If vector A is multiplied by a negative scalar quantity -m, then the product -mA is directed opposite A and has magnitude mA.

F

2F

- 2F

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Quick Quiz If vector B is added to vector A,

under what condition does the resultant vector A + B have magnitude A + B? Under what conditions is the resultant vector equal to zero?

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Homework See online homework assignment

at www.masteringphysics.com