There are different laws of vector addition and they are

A vector is a physical quantity which is represented both in direction and magnitude.

Suppose, we have two vectorsand B as shown.

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.

Now after this join the other end points of both the vectors together as shown,

The resultant of the given vectors is given by vector C which 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

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

Now we can mathematically represent this as

C = A – B

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

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.

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 = $$a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k}$$ and b = $$b_1 \hat{i} + b_2 \hat{j} + b_3 \hat{k}$$ is given by adding the components of the three axes separately.

i.e. a + b = $$a_i \hat{i} + a_2 \hat{j} + a_3 \hat{k} + b_1 \hat{i} + b_2 \hat{j} + b_3 \hat{k}$$

$$\Rightarrow a + b$$ = $$(a_1 +b_1)\hat{i} + (a_2 + b_2)\hat{j} + (a_3 + b_3)\hat{k}$$
Similarly the difference can be given as $$a – b$$ = $$(a_1 – b_1)\hat{i} + (a_2 – b_2)\hat{j} + (a_3 – b_3) \hat{k}$$

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

Example: Let $$\overrightarrow{a}$$ =$$3\hat{i} + 4\hat{j} – 7\hat{k}$$ and $$\overrightarrow{b}$$ =$$6\hat{i} + 4\hat{j} – 6\hat{k}$$. Add both the vectors.

Solution: As both the vectors are already expressed in co-ordinate system we can directly add these as follows
$$\overrightarrow {a} + \overrightarrow{b}$$ =$$(3 + 6)\hat{i} + (4 + 4)\hat{j} + (-7 – 6)\hat{k}$$
or $$\overrightarrow{a} + \overrightarrow{b}$$ =$$9\hat{i} + 8\hat{j} -13\hat{k}$$