Equality of vectors
Addition of vectors
Magnitude of av+bv = SQRT(a²+b²+ 2ab cos θ)
The angle of the resultant with av is α where
tanα = b sin θ/(a+b cos θ)
Interesting point to make note of:
Two vectors having equal magnitudes of a make an angle θ with each other. Find the magnitude and direction of the resultant(Resultant is output of the addition of two vectors.)
Magnitude = 2a cos θ/2
tan α = a sin θ/(a + a cos θ) = (2asin(θ/2)cos(θ/2))/(2acos²(θ/2))
= tan (θ/2)
Example: Two vectors are of equal magnitude of 10 units. One of them is inclined at 45° to the X-axis and the other is inclined at 75° to the X-axis. Find the magnitude and direction of the resultant with respect to X-axis.
The angle between vectors is 30°.
Hence magnitude of the resultant will be 20 cos 15°
The direction - The resultant is inclined at 60° to the X axis.
Subtraction of vectors
Multiplication of vector by a number
2.6 Resolution of vectors
2.7 Dot product or scalar of two vectors
2.8 Cross product or vector product of two vectors
2.9 Differential Calculus
Concepts form Calculus
dy/dx as rate measure
2.10 Maxima and minima
2.11 Integral Calculus
2.12 Significant digits
2.14 Errors in measurement
In recording measurements in experiments, several errors can occur. The equipment can be set in a faulty manner, and experimenter can make errors. These errors can be corrected by supervisors and trainers. But still some errors are committed due to random noncontrollable causes.
As random errors are sometime positive and sometimes negative, average is considered as true value. But sigma or standard deviation of the measurements can be calculated and 1.96 sigma limits and 3 sigma limits can be determined to specify confidence limits for true value. But at least 8 measurements are to be taken to get a good statistical estimate for average and standard deviation.
Update 20 May 2015
Earlier update 7 May 2008