Resolution of Vectors

If + = , the resultant, then conversely = + i.e. the vector can be split up so that the vector sum of the split parts equals the original vector . If the split parts are mutually perpendicular then they are known as components of and this process is known as resolution. The orthogonal component of any vector along another direction which is at an angular separation θ is the product of the magnitude of the vector and cosine of the angle between them (θ). Therefore the component of is A cosθ.

Note: In physics, resolution gives unique and mutually independent components only if the resolved components are mutually perpendicular to each other. Such a resolution is known as rectangular or orthogonal resolution and the components are called rectangular or orthogonal components.

Unit vector along the direction of is = /A, Where A is magnitude of . are the unit vectors along positive direction of X, Y and Z axis respectively, then the rectangular resolution of vector can be represented. where AX, AY, AZ are the magnitudes of X, Y and Z components of . The magnitude of vector is given by .

Illustration 8. A force of 30 N is acting at an angle of 600 with the y-axis. Determine the components of the forces along x and y-axes.

Illustration 9.Four forces act along the sides of a smooth square frame ABCD in the order A → B, B → C, C →D and D → A. If the magnitude of the forces are F1, F2, F3 and F4 respectively, find the resultant force acting on the frame. Assume F ₁ = 1 N, F ₂ = 2N, F ₃ = 3N and F ₄ = 4N.

Solution :Let us consider x – y coordinate system. The resultant of all the forces is ;

After bringing the tails of all the vectors to a point O and substituting ,

We have,

Hence is directed in 3rd quardrant as shown in the figure.