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Now that we have learned to break a vector into components, we can begin adding vectors using components.

The wonderful thing about vector components is that once we chose a coordinate system, all x-components of vectors point in the same direction.  This means that we can add the x-components of two vectors by simply adding them.  The same holds true for the y-components.

In order to add two random vectors, we simply break each into components.  We then add the x-components together.  Then we add the y-components together.  Finally, we use the Pythagorean theorem to find the resultant and the trig functions to find the direction.

You can review finding components and adding perpendicular vectors.

This animated .gif outlines the process.

We first break the two vectors,  $$\color{black}{\vec{A}}$$ and $$\color{black}{\vec{B}}$$ into components.

$$\color{black}{\vec{A} = \vec{A_x} + \vec{A_y}}$$ and $$\color{black}{\vec{B} = \vec{B_x} + \vec{B_y}}$$

It might help to write these as

$$\color{black}{\vec{A} = A_x \hat{x} + A_y \hat{y}}$$ and $$\color{black}{\vec{B} = B_x \hat{x} + B_y \hat{y}}$$.

The x-component of the resulting vector $$\color{black}{\vec{C}}$$ is just the sum of the x-components of $$\color{black}{\vec{A}}$$ and $$\color{black}{\vec{B}}$$

$$\color{black}{\vec{C_x} = A_x \hat{x} + B_x\hat{x}}$$.

Likewise, the y- component is given by

$$\color{black}{\vec{C_y} = A_y \hat{y} + B_y\hat{y}}$$.

We could write this as

$$\color{black}{\vec{C} = (A_x + B_x) \hat{x} + (A_y + B_y) \hat{y}}$$.

Because the x-components are parallel, we could even go so far as to just write

$$\color{black}{C_x = A_x + B_x}$$ and $$\color{black}{C_y = A_y + B_y}$$.

Once we have the values of $$\color{black}{C_x}$$ and $$\color{black}{C_y}$$, we use the Pythagorean theorem to find C…

$$\color{black}{C^2 = C^2_x + C^2_y}$$.

Finally, we use the tangent function to find $$\color{black}{\theta_c}$$…

$$\color{black}{\tan \theta = \frac{C_y}{C_x}}$$.