Chapter 3 Vectors. Coordinate Systems Used to describe the ___________of a point in space Coordinate...
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Transcript of Chapter 3 Vectors. Coordinate Systems Used to describe the ___________of a point in space Coordinate...
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.