a. If \vec{v} + \vec{t} =\vec{v}, what is \vec{t}?
b. If t\vec{v} = \vec{v}, what is t?
c. If s\vec{v} = t\vec{u}, and \vec{u} is not parallel to \vec{v}, what are s and t?
Using vector diagrams, show that
\displaystyle
(a+ b) \vec{u} = a\vec{u} + b\vec{u}
Using vector diagrams, show that
\displaystyle
(a b) \vec{u} = a(b\vec{u})
A mass M is hung on a line between two supports A and B.
a. Which part of the line supporting the mass has the greater tension? Explain.
b. The supports A and B are not at the same level. Whats effect does this have on the tension in the line? Explain.
If \vec{i} and \vec{j} are perpendicular unit vectors, what is the magnitude of
\displaystyle
3\vec{i} + 4\vec{j}
If \vec{i} and \vec{j} are perpendicular unit vectors, what is the magnitude of
\displaystyle
24\vec{i} - 7\vec{j}
If \vec{i} and \vec{j} are perpendicular unit vectors, what is the magnitude of
\displaystyle
a\vec{i} + b\vec{j}
Show that |\vec{a}| + |\vec{b}| = |\vec{a} -\vec{b}|, if \vec{a} and \vec{b} have opposite directions.
A 3-kg mass is hanging from the ned of a string. If a horizontal force of 12 N pulls the mass to the side
a. find the tension in the string
b. find the angle the string makes with the vertical
Two forces \vec{F_1} and \vec{F_2} act on an object. Determine the magnitude of the resultant if
|\vec{F_1}| = 54N, |\vec{F_2}| = 34N, and the angle between them is 55^o
Two forces \vec{F_1} and \vec{F_2} act on an object. Determine the magnitude of the resultant if
|\vec{F_1}| = 21N, |\vec{F_2}| = 45N, and the angle between them is 140^o
Two forces at an angle of 130^o to each other act on an object. Determine their magnitudes if the resultant has a magnitude of 480 N and makes an angle of 55^o with one of the forces.
Forces of 5N, 2N, and 12N, all lying in the same plane, act on an object. The 5 N and 2N forces lie on opposite sides of the 12 N force at angles of 40^o and 20^o, respectively. Find the magnitude and direction of the resultant.
A 10-kg mass is supported by two strings of length 5m and 7 m attached to two points in the ceiling 10 m apart. Find the tension in each string.
The pilot of an airplane that flies at 800 km/h wishes to travel to a city 800 km due east. There is a 80 km/h wind from the northeast.
a. What should the plane's heading be?
b. How long will the trip take?
An airplane heads due south with an air speed of 480 km/h. Measurements made from the ground indicate that the plane's ground speed is 528 km/h at 15^o east of south. Calculate the wind speed.
A camp counsellor leaves a dock paddling a canoe at 3 m/s. She heads downstream at 30^o to the current, which is flowing at 4 m/s.
a. How far downstream does she travel in 10 s?
b. What is the length of time required to cross the river if its width is 150 m?
A pilot wishes to reach an airport 350 km from his present position at a heading of N 60^oE. If the wind is from S25^oE with a speed of 73 km/h, and the plane has an airspeed of 450km/h, find
a. what heading the pilot should steer
b. what the ground speed of the plane will be
c. how many minutes it will take for the plane to reach its destination
A coast guard cutter is steering west at 12 knots, when its radar detects a tanker ahead at a distance of 9 nautical miles travelling with a relative velocity of 19 knots, on a heading of E14^oN. What is the actual velocity of the tanker?
Twice a week, a cruise ship carries vacationers from Miami, Florida to Freeport in the Bahamas, and then on the Nassau before returning to Miami. The distance form Miami t oFreeport is 173 km on a heading of E20^oN. The distance from Freeport to Nassau is 217 km on a heading of E50^oS. Once a week the ship travels directly from Miami to Nassau. Determine the displacement vector from Miami to Nassau.
If a\vec{u} + b\vec{v} = \vec{0} and \vec{u} and \vec{v} have different directions, what must a and b equal?
Show geometrically that \vert \vert \vec{u} \vert - \vert \vec{v} \vert \vert \leq \vert \vec{u} + \vec{v} \vert. Under what conditions does equality hold?