and are unit vectors along x- and y- axis respectively. What is the magnitude and direction of the vectors, and ? What are the components of a vector along the directions of and? [You may use graphical method]

Any vector quantity in x-y plane is represented as linear combination of two unit vectors one along x direction and other along y direction namely and , coefficient of and represent magnitude of vector quantity in x and y direction respectively

for any vector

of p_x and p_y represent magnitude of vector quantity in x and y direction respectively

the magnitude is given by

And angle with x axis is given by

Here let the vectors be

and

Now considering

X component a_x = 1 unit

Y component a_y = 1 unit

So magnitude of vector is

Or

Now for direction angle with x axis

Or

So angle made with x axis is

Now considering

X component b_x = 1 unit

Y component b_y = -1 unit

So magnitude of vector is

Or

Now for direction angle with x axis

Or

So angle made with x axis is

As shown Graphically in the figure

Now component of one vector in the direction of other vector is the magnitude of vector in direction of other

Now let us say we have to find component of in the direction of let us angle between them is