Physics 170 - Mechanics Lecture 2 Vectors

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Scalar and Vector QuantitiesScalar quantities are completely described by magnitude only (temperature …).

Vector quantities need both magnitude (size) and direction to completely describe them (force, displacement, velocity,…).

Represented by an arrow, the length of the arrow is proportional to the magnitude of the vector.

Head of the arrow represents the direction.

Indicated by the name with an arrow on top of it.

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Scalars Versus Vectors

How to get to the Library: you need to know how far (0.5 miles) and which way.

Need to have consistent systems of units for the measurements

Need rules for dealing with the uncertainties

Scalar: quantity is a number with units

Vector: quantity has both a magnitude (with units) and a direction

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The Components of a Vector

Even though you know how far and in which direction the library is, you may not be able to walk there in a straight line.

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Math Review: Trigonometry

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Example: how high is the building?

• Known: angle and one side • Find: !another side • Key: tangent is defined via • two sides!

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• Length, angle, and components can be calculated from each other using trigonometry:

• Convention: We measure counterclockwise from the +x axis

The Components of a Vector

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2D Cartesian and Polar Coordinate Representations

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The Components of a Vector

• We can resolve vector into perpendicular components using two-dimensional coordinate systems:

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Example: Height of a Cliff In Jules Vern’s Mysterious Island,

Capt. Cyrus Harding wants to find the

height of a cliff. He stands with his

back to the base of the cliff and

marches straight away from it for 500 ft.

At this point, he lies on the ground and

measures the angle from horizontal to

the top of the cliff. (a) If the angle is 34.0°, how high is the cliff?

(b) What is the straight line distance d from Capt. Harding to the top of the cliff?

Any valid physical equation must be dimensionally 10

The Components of a Vector

• Signs of vector components and range of :

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Question 1A vector has a negative x-component and a positive y-component. The angle theta that specifies its direction is measured counterclockwise from the x axis. In what range is theta ?

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Combining Vectors Graphically

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Adding Vectors

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Adding Vectors Graphically

Adding vectors graphically:

Place the tail of the second at the head of the first. The sum points from the tail of the first to the head of the last.

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Subtracting Vectors

The negative of a vector is a vector of the same magnitude pointing in the opposite direction. Here, D = A – B.

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Subtracting Vectors

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Adding Vectors by Components

Adding Vectors by Components

• 1) Find the components of each vector to be added.

• 2) Add the x- and y-components separately.

• 3) Find the resultant vector.

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Multiplication by a Scalar

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Properties of Vectors

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• Vectors are equal if they have the same magnitude and direction.

Properties of Vectors

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Vector Summary

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Graphical vector addition: Place tail of second at head of first; sum points from tail of first to head of last or parallelogram method placing vectors to the same origin.

Unit VectorsUnit vectors are dimensionless vectors of unit length. For example, the vector

has magnitude (with units) Ax and points along the +x axis.

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Unit Vectors

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Sum of vectors and product with a scalar

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