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Vector Subtraction

Vector Subtraction in 2D

Vector Subtraction involves calculating the difference of vectors. This is done by subtracting 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 difference of the vectors.

Below is a simulation of vector subtraction in 2D. There is a red vector of magnitude 8.6, and an angle of -35.54 degrees. This vector is subtracted by a blue vector of variable magnitude and angle because the vector is dependent upon the movement of your mouse over it. The x and y components of the blue vector are subtracted from the x and y components of the red vector. The new vector, which is purple, now has a magnitude based on the x and y axis components of the red vector minus 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 difference of the red and blue vectors.

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

\[ \vec{A} = \begin{bmatrix} \hat{i} & \hat{j} & \hat{k} \\ 7 & -5 & 0 \end{bmatrix} \]

or

\[ \vec{A} = 7\hat{i} - 5\hat{j} + 0\hat{k} \]

or

\[ \vec{A} = \langle 7, -5, 0 \rangle \]

and

\[ \vec{B} = \begin{bmatrix} \hat{i} & \hat{j} & \hat{k} \\ -9 & -12 & 0 \end{bmatrix} \]

or

\[ \vec{B} = -9\hat{i} - 12\hat{j} + 0\hat{k} \]

or

\[ \vec{B} = \langle -9, -12, 0 \rangle \]

Vector Subtraction can be calculated using the following equations:

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

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 subtract the red and blue vectors:

\[ \vec{A} = 7\hat{i} - 5\hat{j} + 0\hat{k} \]

\[ \vec{B} = -9\hat{i} - 12\hat{j} + 0\hat{k} \]

\[ \vec{A} - \vec{B} = \langle 7 - -9, -5 - -12, 0 - 0 \rangle \]

\[ \vec{A} - \vec{B} = \langle 16, 7, 0 \rangle \]

\[ \vec{C} = \langle 16, 7, 0 \rangle \]

or

\[ \vec{A} - \vec{B} = (7 - -9)\hat{i} + (-5 - -12)\hat{j} + (0 - 0)\hat{k} \]

\[ \vec{A} - \vec{B} = (16)\hat{i} + (7)\hat{j} + (0)\hat{k} \]

\[ \vec{C} = 16\hat{i} + 7\hat{j} + 0\hat{k} \]

As we can see from the formula, when we subtract 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 difference of the red and blue vectors.

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