Solving Problems with Vectors

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


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:


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 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:


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


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.