Vector Addition by Components is very simple. To add two vectors, first resolve the vectors into x- and y-components, and add the components.

So, if you want to add $$\vec{A}$$ and $$\vec{B}$$, first write

$$\vec{A} = A_x \hat{i} + A_y \hat{j}$$
$$\vec{B} = B_x \hat{i} + B_y \hat{j}$$

Then, the new vector

$$\vec{C} = C_x \hat{i} + C_y \hat{j}$$

will have

$$C_x = A_x + B_x$$
$$C_y = A_y + B_y$$