Rectangular and polars
Can the magnitude of a vector have a negative value?
Explain.
No, the magnitude of a vector is always positive. A minus sign in a vector only indicates direction, not magnitude.
Vector A=has a magnitude of 35.0 units and points in the direction 325° counterclockwise from the positive x axis. Calculate the x and y components of this vector.
Your dog is running around the grass in your backyard. He undergoes successive displacements 3.50 m south, 8.20 m northeast, and 15.0 m west. What is the resultant displacement?
R=9.48m
Obtain expressions in component form for the position vectors having the polar coordinates (a) 12.8 m, 150°; (b) 3.30 cm, 60.0°; and (c) 22.0 in., 215°.
a) (-11.1, 6.40) m
b) (1.65, 2.86) m
c) (-18, -12.6) m
The polar coordinates of a point are r=5 5.50 m and a=240°. What are the Cartesian coordinates of this point? (a is the direction)
x=-2.75
y=-4.76
A car travels 20.0 km due north and then 35.0 km in a direction 60.0° west of north. Find the magnitude and direction of the car’s resultant displacement.
What is R=48.2 km and b=39°
Two points in the xy plane have Cartesian coordinates (2.00, 24.00) m and (23.00, 3.00) m. Determine (a) the distance between these points and (b) their polar coordinates.
a) 8.60 m
b) r1=4.47 m a1=-63.4°; r2=4.24m a2=135°
The rectangular coordinates of a point are given by (2, y), and its polar coordinates are (r, 30°). Determine (a) the value of y and (b) the value of r.
a) 2.31
b) 1.15
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 R for the trip. Find an expression for R in terms of unit vectors.
Rx=37.7 km; Ry=16.9km
R=16.9 km
R=(37.7 i + 16.9 j)
Find the sum of two vectors A and B lying in the xy plane and given by
A=(2.0i + 2.0j) m and B = (2.0i - 4.0j) m
Rx = 4.0 m Ry = -2.0 m
R=4.5 m
A book is moved once around the perimeter of a tabletop with the dimensions 1.0 m by 2.0 m. The book ends up at its initial position. (a) What is its displacement? (b) What is the distance travelled?
(a) The book’s displacement is zero, as it ends up at the point from which it started. (b) The distance travelled is 6.0 meters.
A commuter airplane takes the route shown in Figure 3.20. First, it flies from the origin of the coordinate system shown to city A, located 175 km in a direction 30.0° north of east. Next, it flies 153 km 20.0° west of north to city B. Finally, it flies 195 km due west to city C. Find the location of city C relative to the origin.
Rx=-95.3km; Ry=232km
A particle undergoes three consecutive displacements:
d1=(15i+30j+12) cm, d2=(23i-14j-5.0k) cm
and d3=(-13i+15j) cm. Find the components of the resultant displacement and its magnitude.
What is...
(25i+31j+7.0k) cm
R=40cm
A force F1 of magnitude 6.00 units acts on an object at the origin in a direction a1=30.0° above the positive x-axis. A second force F2 of magnitude 5.00 units acts on the object in the direction of the positive y-axis. Find the magnitude and direction of the resultant force F1 + F2.
a=57°
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 displacement for each day.
Ax=17.7 km; Bx= 20 km
Ay= -17.7 km; By= 34.6 km