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 _________ Cartesian Polar

Polar to Cartesian Coordinates Based on forming a right triangle from r and  x = ________ y = ________

Cartesian to Polar Coordinates r is the hypotenuse and  an angle ____________________  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

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 A x is the same as ______ and A y is the same as _________etc. The complete vector can be expressed as: __________________

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.