How to resolve a vector in its components

That's a bit of a vague question, but…

Typically, when someone asks this, they are dealing with introductory physics or similar and are trying to solve something in two dimensions. In that case, the answer is to break each vector into its X and Y components.

If you don't have a coordinate system given, orient yours such that as many vectors as possible lie along the X or Y axis (the answer won't change, but this makes the math easier)

Make a table, one row for each vector, with 3 columns: vector, X component, Y component

Use cosine and sin functions to break each vector into its components.

Sum the X and the Y columns, the totals are the components of the net (or resultant) vector.

The magnitude of the resultant vector is given by the Pythagorean formula. The angle by inverse tangent of Y/X.

Suppose a vector has magnitude r and angle t w.r.t. x axis

Then x component of vector is r.cos(t)

And y component is r.sin(t)

Hence our vector is r.cos(t) i + r.sin(t) j

Much easier to explain in person with a whiteboard, but hopefully this is of some help.

Hi Harsh

The parallelogram method of vector resolution involves using an accurately drawn, scaled vector diagram to determine the components of the vector. Briefly put, the method involves drawing the vector to scale in the indicated direction, sketching a parallelogram around the vector such that the vector is the diagonal of the parallelogram, and determining the magnitude of the components (the sides of the parallelogram) using the scale. If one desires to determine the components as directed along the traditional x- and y-coordinate axes, then the parallelogram is a rectangle with sides that stretch vertically and horizontally. A step-by-step procedure for using the parallelogram method of vector resolution is:

1. Select a scale and accurately draw the vector to scale in the indicated direction.
2. Sketch a parallelogram around the vector: beginning at the tail of the vector, sketch vertical and horizontal lines; then sketch horizontal and vertical lines at the head of the vector; the sketched lines will meet to form a rectangle (a special case of a parallelogram).
3. Draw the components of the vector. The components are the sides of the parallelogram. The tail of the components start at the tail of the vector and stretches along the axes to the nearest corner of the parallelogram. Be sure to place arrowheads on these components to indicate their direction (up, down, left, right).
4. Meaningfully label the components of the vectors with symbols to indicate which component represents which side. A northward force component might be labeled F north . A rightward velocity component might be labeled v x ; etc.
5. Measure the length of the sides of the parallelogram and use the scale to determine the magnitude of the components in real units. Label the magnitude on the diagram.

The step-by-step procedure above is illustrated in the diagram below to show how a velocity vector with a magnitude of 50 m/s and a direction of 60 degrees above the horizontal may be resolved into two components. The diagram shows that the vector is first drawn to scale in the indicated direction; a parallelogram is sketched about the vector; the components are labeled on the diagram; and the result of measuring the length of the vector components and converting to m/s using the scale.

You can find more on http://www.physicsclassroom.com/class/vectors/Lesson-1/Vector-Resolution

All the Best. :)