Vector addition - Solved problem

A force of 4N is included at an angle of 45 degree to a second force of 7 N, both forces acting at a point.
Find the magnitude of the resultant of these two forces and the direction of the resultant with respect to the 7N force by both the ‘triangle’ and the ‘parallelogram’ methods.

Solution

The forces are shown in the Fig (a). Although the 7N force is shown as a horizontal line, it could have been drawn in any direction.

 nose-to-tail method - Fig b

Using the ‘nose-to-tail’ method, a line 7 units long is drawn horizontally to give vector oa in Fig. (b).
To the nose of this vector ar is drawn 4 units long at an angle of 45 degree to oa. The resultant of vector addition is or and by measurement is 10.2 units long and at an angle of 16◦ to the 7N force.


parallelogram method - Fig c

Figure (c) uses the ‘parallelogram’ method in which lines are drawn parallel to the 7N and 4N forces from the noses of the 4N and 7N forces, respectively.
These intersect at R. Vector OR gives the magnitude and direction of the resultant of vector addition and as obtained by the ‘nose-to-tail’ method is 10.2 units long at an angle of 16 degree to the 7N force.

Resolution of vectors

A vector can be resolved into two component parts such that the vector addition of the component parts is equal to the original vector. The two components usually taken are a horizontal component and a vertical component.
For the vector shown as F in the Figure, the horizontal component of F is Fcosθ and the vertical component is Fsinθ.

the horizontal component of F is Fcosθ and the vertical component is Fsinθ.

Example 1:

Resolve the force vector of 30 Newton at an angle of 35 Degree to the horizontal into a horizontal and a vertical component

Solution

Resolving force vector of 30 Newton at an angle of 35 Degree to the horizontal into a horizontal and a vertical component

  

Example 2:

Resolve the acceleration vector of 17 m/s2 at an angle of 120◦ to the horizontal into a horizontal and a vertical component

Solution

Horizontal component    ,
 acting from left to right.

Vertical component ,
acting vertically upwards.

These component vectors are shown in the Figure: