Dot product

A dot product is an operation that which involves multiplication of two vectors to arrive to a scalar product.  Given two vectors, v=ai + bj and u=ci + dj, v.u read as ‘v dot u’ would be equal to a scalar product, ac + bd. So, basically the product would be a number and not a vector.
The dot product of two vectors would be a scalar even in a three dimensional space, R3.  So, in a three dimensional space given vectors v=ai +bj+ck and u=xi+yj+zk, the dot-product is given by v.w=ax + by +cz. The definition of dot product can be given as the dot product equation of vectors a’ and b’ such that a.b= ax. bx + ay.by = |a||b|cos(theta) .
Here |a| and |b| are the magnitudes of the vectors and theta is the angle between the vectors. It is read as modulus of vector a multiplied with the modulus of vector b, multiplied by the cosine of the angle between the two vectors a’ and b’.
Following are some of the important points to be remembered while finding the scalar product, i.i=1, j.j=1, k.k=1, i.j=0, j.k=0 and k.i=0, this shows that the scalar product of vectors which are perpendicular to each other is zero.
Some of the properties of dot-product are as given below,
• Commutative property: u.v = v.u
• Distributive property: u.(v+w) = (u.v) + (u.w)
• Associative property: (cv). u = v.(cu)= c(u. v)
• 0. u = u.0 = 0
• v.v =|v|2
• If v. v = 0 then v = 0
Let us now take a look at the dot product proof of distributive property given by u. (v+w)=(u.v)+(u.w)
Let the vectors to be, u=(u_1,u_2,u_3...,u_n ); v =(v_1,v_2,v_3...,v_n) and w=(w_1,w_2,w_3...,w_n). On the left hand side we have, u.(v+w) = (u_1,u_2,u_3...,u_n ).[(v_1,v_2,v_3...,v_n)+ (w_1,w_2,w_3...,w_n)]
          =  (u_1,u_2,u_3...,u_n ).[(v_1+w_1), (v_2+w_2), (v_3+w_3)…, (v_n+w_n)] on regrouping we get,
          = [u_1((v_1+w_1), u_2(v_2+w_2), u_3(v_3+w_3),…,u_n  (v_n+w_n)]
Applying the distributive property we get,
= [u_1v_1+u_1w_1, u_2v_2+ u_2w_2, u_3v_3+ u_3w_3….., u_n v_n+ u_n w_n]
Which can be written as, [u_1v_1, u_2v_2, u_3v_3…, u_n v_n] + [u_1w_1, u_2w_2, u_3w_3…, u_n w_n]
On re-writing the above expression we get, [(u_1,u_2,u_3...,u_n ). (v_1,v_2,v_3...,v_n)]+[ (u_1,u_2,u_3...,u_n ). (w_1,w_2,w_3...,w_n)]  which would be the expression on the left hand side, [u.v+u.w] and hence proved!Thus we can prove all the properties using the above computational method.

 u_n it vectors are the vectors with length of one u_n it.  For u_n it vectors u and v, the dot product of u_n it vectors is given by, u.v=cos(theta) where (theta) is the angle between the two u_n it vectors.

Work and Time Calculation

Work and time are two of inter-related concepts in mathematics and science. Work and time related calculations are most often asked in almost all competitive exams. Taught in middle school classes, work and time calculation problems are worked out in SAT, MAT exams as well. The trick is to solve the problems within seconds. Let’s have a look at some of the facts related to work and time calculation in this post.

1. If a person can complete a work in n days, then the person can complete 1/n part of the work in one day. For example: She completed the process of researching, ordering and buying the Fisher Price toys for infants’ collection for her shop in 6 days. Therefore, she will complete 1/6 part of the work of researching, ordering and buying the Fisher Price toys for infants’ collection for her shop.

2. If the number of person to complete a particular work is increased, the time to complete the same work decreases. For example: 100 employees build about 1000 toy action figures in 10 days. If the number of employees is increased to 150, then they will build 1000 toy action figures in less than 10 days because the work is distributed among more workers.

3. If worker A has the capability of working twice as worker B, then A will take ½ of the time that B took to complete a work. For example: B designed the outlook of cot mobile for baby girls in 2 hours. A works twice as B and therefore, A designed the outlook of cot mobile for baby girls in ½ x 2 hours = 1 hour.

These are some of the most important facts to be known while working out work and time calculation in mathematics.  However, the list if not the ultimate one, there are many other such work and time related facts.

Absolute Error

When we do any calculations there are always chances of making mistakes, either we do addition, subtraction or anything, similarly when we measure height, distance or anything with the help of any measuring device there are chances of making a mistake so if we measure the same thing twice we may get different answers and this is due to the error in measuring. Error is not the mistake we have made because it does not give you the wrong answer. The uncertainty in measurement is termed as the error. There are many types of errors which occur in experimental studies.

1. Greatest possible error – This is the error we make when we do the approximation or rounding off to tenth, hundredth place.

2. Absolute Error– This is the error which occurs due to the inaccuracy in the measurement we do. Experimental scientists come across usually with this type of error. This is the amount of physical error we make in the process of measurement. Absolute Error Formula– It is usually denoted by delta x and is equal to difference between the calculated value and the actual value. Now How to Calculate Absolute Error or How to Find Absolute Error– We can find the absolute-error by finding the difference between the inferred value and the calculated value of the measurement. It usually signifies the uncertainty in the measurement process. For example: - If we find the length of stick as 1.09 centimeter though its actual length is 1 centimeter. Then the absolute-error that is delta x = Calculated value – Actual value which is 1.09 – 1 and that is equal to 0.09. Hence we can say that absolute-error is equal to 0.09. Absolute-error is always positive. Therefore we can call it as the absolute value of the difference of the two values which are the calculated value and the actual value.

3. Relative error – This type of error tells you about how good a measurement is relative to size of the thing which is measured. It expresses the ratio of absolute-error to the measurement that is accepted. This actually shows the relative size of the error of the measurement in relation to the measurement itself. The formula for calculating relative error is Relative error = Absolute error over accepted measurement.