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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

\(C_x = A_x + B_x\)
\(C_y = A_y + B_y\)