**Scalars**

**Vectors**

**Equality of vectors**

**Addition of vectors**

Magnitude of a

^{v}+b

^{v}= SQRT(a²+b²+ 2ab cos θ)

The angle of the resultant with a

^{v}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

and

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**

**Zero vector**

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