Chapter 3: Vectors EXAMPLES

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: From Equation 3.4, and from Equation 3.3,

Example 3.1, cont. Change the point in the x-y plane Note its Cartesian coordinates Note its polar coordinates Please insert active fig. 3.3 here

Example 3.2 V = Vector Displacement 500 m, 30º N of E. Find components of V (Vx and Vy )

Example 3.3 Sum of Two vectors (Example 3.3 Text Book) Find the Resultant vector: R = A + B If: and Using Eqn: (3.14) Or: Rx = 4.0m and Ry = – 2.0m Magnitude and direction of R will be: –27o means clockwise from + x axis. Or 333o from +x axis counterclockwise

Example 3.4 Taking a Hike (Example 3.5 Text Book) 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.

Example 3.4 cont, Find the resultant displacement (graphically and analytically) for the trip: R = A + B Select a coordinate system Draw a sketch of the vectors Find the x and y components of A & B (Decomposition) y Bx B By Ax x Ay A

Example 3.4 cont, Draw each component with its magnitude and direction Find Rx and Ry components of the resultant: Rx = Σx components Ry = Σy components Given by Equation 3.15: Rx = Ax + Bx = 17.7 km + 20.0 km Rx = 37.7 km Ry= Ay + By = –17.7 km + 34.6 km Ry = 16.9 km In unit-vector form, we can write the total displacement as y By Ry Bx x Ax Rx Ay

Example 3.4 cont, Draw Rx and Ry components with its magnitude and direction Use the Parallelogram system to find the resultant graphically Use the Pythagorean theorem to find the magnitude of the resultant (R) And the tangent function to find the direction (θ ) y Ry R  x Rx

Example 3.5 Conceptual Questions Q1: Two vectors have unequal magnitudes. Can their sum be Zero? NO! The sum of two vectors are only zero if they are in opposite direction and have the same magnitude!!! Q9: Can the magnitude of a vector have a negative value? The magnitude of a vector is always positive. A negative sign in a vector only means DIRECTION!!!!

Material for the Midterm Material from the book to Study!!! Objective Questions: 3-8-10 Conceptual Questions: 2-3-4 Problems: 6-7-15-23-29-45-57