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Vector Components

Vector Components in 1D x-axis

Vector Components are smaller vectors that help split an angled vector towards the coordinate axes in a two and three dimensional coordinate system.

In the simulation below, you can see a vector of variable magnitude based on the movement of your mouse over the simulation. Notice how the x-axis component length of the vector is positive, when it has an angle of 0, and a direction to the east, whereas the x-axis component length of the vector is negative when it has an angle of 180 and a direction to the west. The numerical value for the magnitude (length) regardless whether or not it is pointing in the positive or negative direction, is always a positive scalar. Notice also how only the x-axis components change, whereas the y-axis component length of the vector is 0. Since the vector is solely on the x-axis, that also means that all other vector components are 0.

A vector and its vector components can be represented in numerous ways. (Note that in this simulation we have positive and negative directions). Here are the three most common ways of portraying them:

\[ \vec{A} = \begin{bmatrix} \hat{i} & \hat{j} & \hat{k} \\ -21 & 0 & 0 \end{bmatrix} \]

or

\[ \vec{A} = -21\hat{i} + 0\hat{j} + 0\hat{k} \]

or

\[ \vec{A} = \langle -21, 0, 0 \rangle \]

and

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

or

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

or

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

Vector Components in the x-axis can be calculated using the following equations:

\[ \cos(\theta) = \frac{\text{Adjacent Side}}{\text{Hypotenuse}} = \frac{v_x}{v} \]

or

\[ v_x = \cos(\theta) \]

Now let's see what happens when we use the numbers chosen:

\[ v_x = 21 \cos(180) \]

\[ v_x = 21 \cdot (-1) \]

\[ v_x = -21 \]

\[ v_x = 5 \cos(0) \]

\[ v_x = 5 \cdot 1 \]

\[ v_x = 5 \]

In the first case, we take the vector to have a magnitude of 21, but pointing in the negative direction, with an angle of 180 degrees. Following the formula, we get that the x-axis component of the vector is -21. Similarly, in the second case, we take the vector to have a magnitude of 5, but pointing in the positive direction, with an angle of 0 degrees. Following the formula, we get that the x-axis component of the vector is 5. Both these results make sense when we look back at the simulation.

The magnitude (length) of the vector once you know the vector components of the vector can be calculated by the following equation:

\[ |\vec{v}| = \sqrt{v_x^2 + v_y^2} \]

Let's calculate the magnitude of the vector in the simulation:

\[ |\vec{v}| = \sqrt{-21^2 + 0^2} \]

\[ |\vec{v}| = \sqrt{441 + 0} \]

\[ |\vec{v}| = \sqrt{441} \]

\[ |\vec{v}| = 21 \]

\[ |\vec{v}| = \sqrt{5^2 + 0^2} \]

\[ |\vec{v}| = \sqrt{25 + 0} \]

\[ |\vec{v}| = \sqrt{25} \]

\[ |\vec{v}| = 5 \]

As we can see from the formula, regardless if a vector component is positive or negative, the component number value gets squared, making it a positive number. Solving the rest of the equation, we find the magnitude, is a positive scalar, which is 21 and 5 above, which is correct.

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