Chapter 3

Vectors

Coordinate Systems

• Used to describe the ___________of a point in space

• Coordinate system consists of– A fixed _____________point called the origin– Specific axes with scales and labels

Cartesian vs Polar Coordinate Systems

– Origin and reference line are noted– Point is distance ________from the

origin in the __________of angle , ccw from reference line

– Points are labeled _________

CartesianPolar

Polar to Cartesian Coordinates

• Based on forming a right triangle from r and q

• x = ________• y = ________

Cartesian to Polar Coordinates

• r is the hypotenuse and q an angle

____________________

____________________• q must be ccw from positive x

axis for these equations to be valid

Example 3.1• The Cartesian coordinates of a

point in the xy plane are (x,y) = (-3.50, -2.50) m, as shown in the figure. Find the polar coordinates of this point.

Solution:

and,

Vector Notation

• Text uses _____with arrow to denote a vector: • Also used for ________is simple bold print: A• When dealing with just the magnitude of a vector in

print, an italic letter will be used: A or _______

– The magnitude of the vector has physical units– The magnitude of a vector is always a __________number

• When handwritten, use an arrow: _____

Adding Vectors

• When adding vectors, their ____________must be taken into account

• ________must be the same • Graphical Methods

– Use _________drawings• Algebraic Methods

– More convenient

To Add Vector’s Graphically

A

B

R

BA

R

Graphical Addition

• Create a _________• Draw the vectors based on the scale

– Put the ______of one vector on the _____of the other

– The resultant vector is the one that goes from the ________of the first vector to the _______of the second.

– Use a protractor, ruler, and your established scale to get the value of the resultant vector.

Problem

at 30° above the x-axis

at 20° below the x-axis

Find graphically.

Multiplying or Dividing a Vector by a Scalar

• The result of the multiplication or division of a vector by a scalar is a _______

• The magnitude of the vector is multiplied or divided by the scalar

• If the scalar is positive, the direction of the result is the _________as of the original vector

• If the scalar is negative, the direction of the result is ____________that of the original vector

Problem

at 30° above the x-axis

at 20° below the x-axis

Find graphically.

Component Method of Adding Vectors

• Graphical addition is not recommended when– High __________is required– If you have a _______________problem

• ___________method is an alternative method– It uses ______________of vectors along

coordinate axes– It gives exact answers

Problem

at 30° above the x-axis

at 20° below the x-axis

Find by components.

Rules of Adding Vectors

• Commutative law:

• Associative law:

• Vector subtraction:

• Algebra still works:

Unit Vectors

• A unit vector is a ____________vector with a magnitude of exactly ____.

• Unit vectors are used to specify a ___________and have no other physical significance

Unit Vectors, cont.

• The symbols______________ represent unit vectors• They form a set of mutually

__________vectors in a right-handed coordinate system

• Remember,

___________________

Unit Vectors in Vector Notation

• Ax is the same as ______

and Ay is the same as _________etc.

• The complete vector can be expressed as:

__________________

j

Problem

at 30° above the x-axis

at 20° below the x-axis

Represent each vector using unit vector notation. Represent the resultant vector of as a unit vector.

Example 3.5 – Taking a Hike• A hiker begins a trip by first walking 25.0 km southeast from her car. She

stops and sets up her tent for the night. On the second day, she walks 40.0 km in a direction 60.0° north of east, at which point she discovers a forest ranger’s tower. Determine the components of the hiker’s resultant displacement for the trip. Find an expression for it in terms of unit vectors.