Solving Problems with Vectors

We can use vectors to solve many problems involving physical quantities such as velocity, speed, weight, work and so on.

Velocity:

The velocity of moving object is modeled by a vector whose direction is the direction of motion and whose magnitude is the speed.

Example :

A ball is thrown with an initial velocity of 70 feet per second., at an angle of 35° with the horizontal. Find the vertical and horizontal components of the velocity.

Let v represent the velocity and use the given information to write v in unit vector form:

v=70( cos(35°) )i+70( sin(35°) )j

Simplify the scalars, we get:

v57.34i+40.15j

Since the scalars are the horizontal and vertical components of v ,

Therefore, the horizontal component is 57.34 feet per second and the vertical component is 40.15 feet per second.

Force:

Force is also represented by vector. If several forces are acting on an object, the resultant force experienced by the object is the vector sum of these forces.

Example :

Two forces F 1 and F 2 with magnitudes 20 and 30lb , respectively, act on an object at a point P as shown. Find the resultant forces acting at P .

First we write F 1 and F 2 in component form:

v57.34i+40.15j

Simplify the scalars, we get:

F 1 =( 20cos( 45° ) )i+( 20sin( 45° ) )j =20( 2 2 )i+20( 2 2 )j =10 2 i+10 2 j F 2 =( 30cos( 150° ) )i+( 30sin( 150° ) )j =30( 3 2 )i+30( 1 2 )j =15 3 i+15j

So, the resultant force F is

F= F 1 + F 2 =( 10 2 i+10 2 j )+( 15 3 i+15j ) =( 10 2 15 3 )i+( 10 2 +15 )j 12i+29j

Work:

The work W done by a force F in moving along a vector D is W=FD .

Example :

A force is given by the vector F= 2,3 and moves an object from the point (1,3) to the point (5,9) . Find the work done.

First we find the Displacement.

The displacement vector is

D= 51,93 = 4,6 .

By using the formula, the work done is

W=FD= 2,3 4,6 =26

If the unit of force is pounds and the distance is measured in feet, then the work done is 26 ft-lb.