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Vector Addition

Vector Addition in 2D

Vector Addition involves calculating the sum of vectors. This is done by adding together the vector components of each individual vector and once the result is obtained, those components are now the components of the new vector that is equal to the sum of the vectors.

Below is a simulation of vector addition in 2D. There is a red vector of variable magnitude and angle, because this vector is dependent upon the movement of your mouse over the purple sum vector. This vector is added to a blue vector of magnitude 10, however at an angle of -53.13 degrees. The x and y components of the red vector are added to the x and y components of the blue vector which are -6 for the x-axis and -8 for the y-axis. The new vector, which is purple, now has a magnitude based on the x and y axis components of the red vector plus the x and y axis components of the blue vector. The purple vector, as you drag your mouse, therefore has a variable magnitude and angle based on the sum of the blue and red vectors.

The vectors in the simulation can be represented in numerous ways. Here are the three most common ways of portraying them:

\[ \vec{A} = \begin{bmatrix} \hat{i} & \hat{j} & \hat{k} \\ -6 & 0 & 0 \end{bmatrix} \]

or

\[ \vec{A} = -6\hat{i} + 0\hat{j} + 0\hat{k} \]

or

\[ \vec{A} = \langle -6, 0, 0 \rangle \]

and

\[ \vec{B} = \begin{bmatrix} \hat{i} & \hat{j} & \hat{k} \\ 0 & -8 & 0 \end{bmatrix} \]

or

\[ \vec{B} = 0\hat{i} - 8\hat{j} + 0\hat{k} \]

or

\[ \vec{B} = \langle 0, -8, 0 \rangle \]

Vector Addition can be calculated using the following equations:

\[ \vec{A} + \vec{B} = \langle A_x + B_x, A_y + B_y, A_z + B_z \]

or

\[ \vec{A} + \vec{B} = (A_x + B_x)\hat{i} + (A_y + B_y)\hat{j} + (A_z + B_z)\hat{k} \]

Now let's see what happens when we add the red and blue vectors:

\[ \vec{A} = -6\hat{i} + 0\hat{j} + 0\hat{k} \]

\[ \vec{B} = 0\hat{i} - 8\hat{j} + 0\hat{k} \]

\[ \vec{A} + \vec{B} = \langle -6 + 0, 0 - 8, 0 + 0 \rangle \]

\[ \vec{A} + \vec{B} = \langle -6, -8, 0 \rangle \]

\[ \vec{C} = \langle -6, -8, 0 \rangle \]

or

\[ \vec{A} + \vec{B} = (-6 + 0)\hat{i} + (0 - 8)\hat{j} + (0 + 0)\hat{k} \]

\[ \vec{A} + \vec{B} = (-6)\hat{i} + (-8)\hat{j} + (0)\hat{k} \]

\[ \vec{C} = -6\hat{i} - 8\hat{j} + 0\hat{k} \]

As we can see from the formula, when we add together the components of the red vector and the blue vector, we get new vector components as a result, which become the vector components for the purple vector, which is the sum of the red and blue vectors.

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