Laws of Vector Addition- Parallelogram, Triangle Law, Vector Addition | Class 12 Math Notes Study Material Download Free PDF

Laws of Vector Addition

A vector is a physical quantity which is represented both in direction and magnitude. In the upcoming discussion, we shall learn about how to add different vectors. There are different laws of vector addition and they are:

  • Triangle law of vector addition
  • Parallelogram law of vector addition

Triangle Law of Vector Addition

Suppose, we have two vectorsand B as shown.

Triangle Law of Vector Addition

Now the method to add these is very simple, what we do is to simply place the head of one vector over the tail of the other vector as shown below.

Triangle Law of Vector Addition 2

Now join the other endpoints of both the vectors together as shown in the below figure.

Triangle Law of Vector Addition 3

The resultant of the given vectors is given by the vector Cwhich represents the sum of vectors  A and B.

i.e. C = A + B

Vector addition is commutative in nature i.e.

if C = A + B; then C = B + A

Or

A + B = C = B + A

Similarly, if you want to subtract both the vectors using the triangle law then simply reverse the direction of any vector and add it to the other one as shown.

Triangle Law of Vector Addition 3

Now, this can be represented mathematically as:

   C = A – B

Parallelogram Law of Vector Addition

This law is also very similar to the triangle law of vector addition. Consider the two vectors again.

Parallelogram Law of Vector Addition

Now for using the parallelogram law, we represent both the vectors as adjacent sides of a parallelogram and then the diagonal emanating from the common point represents the sum or the resultant of the two vectors and the direction of the diagonal gives the direction of the resultant vector.

Parallelogram Law of Vector Addition 2

The resultant vector is shown by C. This is known as the parallelogram law of vector addition.

By using the orthogonal system of vector representation the sum of two vectors

a =

\(\begin{array}{l}a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k}\end{array} \)

 and b =

\(\begin{array}{l}b_1 \hat{i} + b_2 \hat{j} + b_3 \hat{k}\end{array} \)

 is given by adding the components of the three axes separately.

i.e. a + b =

\(\begin{array}{l}a_i \hat{i} + a_2 \hat{j} + a_3 \hat{k} + b_1 \hat{i} + b_2 \hat{j} + b_3 \hat{k} \end{array} \)

\(\begin{array}{l}\Rightarrow a + b \end{array} \)

=

\(\begin{array}{l}(a_1 +b_1)\hat{i} + (a_2 + b_2)\hat{j} + (a_3 + b_3)\hat{k} \end{array} \)


Similarly, the difference can be given as

\(\begin{array}{l}a – b\end{array} \)

=

\(\begin{array}{l}(a_1 – b_1)\hat{i} + (a_2 – b_2)\hat{j} + (a_3 – b_3) \hat{k}\end{array} \)

Now let us take an example to understand this topic better.

Example: Let

\(\begin{array}{l}\overrightarrow{a}\end{array} \)

=

\(\begin{array}{l} 3\hat{i} + 4\hat{j} – 7\hat{k}\end{array} \)

 and

\(\begin{array}{l}\overrightarrow{b}\end{array} \)

=

\(\begin{array}{l} 6\hat{i} + 4\hat{j} – 6\hat{k}\end{array} \)

. Add both the vectors.

Solution: As both the vectors are already expressed in co-ordinate system we can directly add these as follows

\(\begin{array}{l}\overrightarrow {a} + \overrightarrow{b}\end{array} \)

=

\(\begin{array}{l} (3 + 6)\hat{i} + (4 + 4)\hat{j} + (-7 – 6)\hat{k}\end{array} \)


or

\(\begin{array}{l}\overrightarrow{a} + \overrightarrow{b}\end{array} \)

=

\(\begin{array}{l} 9\hat{i} + 8\hat{j} -13\hat{k}\end{array} \)

Er. Neeraj K.Anand is a freelance mentor and writer who specializes in Engineering & Science subjects. Neeraj Anand received a B.Tech degree in Electronics and Communication Engineering from N.I.T Warangal & M.Tech Post Graduation from IETE, New Delhi. He has over 30 years of teaching experience and serves as the Head of Department of ANAND CLASSES. He concentrated all his energy and experiences in academics and subsequently grew up as one of the best mentors in the country for students aspiring for success in competitive examinations. In parallel, he started a Technical Publication "ANAND TECHNICAL PUBLISHERS" in 2002 and Educational Newspaper "NATIONAL EDUCATION NEWS" in 2014 at Jalandhar. Now he is a Director of leading publication "ANAND TECHNICAL PUBLISHERS", "ANAND CLASSES" and "NATIONAL EDUCATION NEWS". He has published more than hundred books in the field of Physics, Mathematics, Computers and Information Technology. Besides this he has written many books to help students prepare for IIT-JEE and AIPMT entrance exams. He is an executive member of the IEEE (Institute of Electrical & Electronics Engineers. USA) and honorary member of many Indian scientific societies such as Institution of Electronics & Telecommunication Engineers, Aeronautical Society of India, Bioinformatics Institute of India, Institution of Engineers. He has got award from American Biographical Institute Board of International Research in the year 2005.