MASTER ADVANCED PHYSICS
(FOR ALL TOP EXAMS)
VECTORS – 4
RESOLUTION OF VECTORS
It is a good idea to break one vector into what is called rectangular components.
What does this mean?
Suppose you have a 2 dimensional vector and we want to have 2 one dimensional vectors, one on the x axis and the other on the y axis.
These two one dimensional vectors sum up and give the original vector.
So, if we have a vector of 10 units and at 60 degrees to the X axis, then
The component on the X axis is A Cos (theta) and the component of the Y axis is A Sin (theta)
Let us check whether this is true.
WE have seen that R2 = a 2 + b2 + 2ab Cos(theta)= a sin2 (theta) + BCos2(theta) + 2ab Cos(theta)
= 10(Sin2(theta) + Cos2(theta) ) + 0 (as Cos 90 is zero
= 10 which is true...
So, we can break a vector into two components along X axis and along Y axis.
If we have say 10 forces acting on a body, and if we break EACH one into components in X axis and y axis, then finally we will have
10 vectors along x axis
10 vectors along y axis.
Each can be added very simply. As it is one dimensional.
And finally we will have after adding, ONE value of the vector along x axis A and one along y axis B.
So, resultant can be got by
R2 = A2 + B2
R = root of a2 + b2
The final angle would be Tan b/a where b= Y component and a= X component.
So these are the basic concepts and also formulae.
PRODUCT OF VECTORS
Dot product – A DOT B = AB COS (THETA)
Cross product – A CROSS B = AB SIN (THETA)
We will see these concepts by linking to physics in later chapters.