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Vector Scaling

Vector Scaling in 1D x-axis

Vector Scaling is the change of a vector's length by a scale factor. Vector Scaling is in other words vector multiplication and division, however these two mathematical operations are combined by the word scaling since if you divide a number by two, that is the same as multiplying a number by one half. Vector Scaling therefore expands or shrinks a vector's length, keeping its angle or inverting it, however.

Below is a simulation of vector scaling in the x-axis. There is a blue vector of magnitude (length) 3, and a second vector, a red vector of variable magnitude (length) based on the movement of your mouse. As you move the red vector, its magnitude (length) is multiplied by a positive or a negative scalar, which changes the magnitude (length) of the vector as well as direction and angle when the scale factor is negative.

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

\[ \vec{A} = \begin{bmatrix} \hat{i} & \hat{j} & \hat{k} \\ 3 & 0 & 0 \end{bmatrix} \]

or

\[ \vec{A} = 3\hat{i} + 0\hat{j} + 0\hat{k} \]

or

\[ \vec{A} = \langle 3, 0, 0 \rangle \]

and

\[ \vec{B} = \begin{bmatrix} \hat{i} & \hat{j} & \hat{k} \\ -3.81 & 0 & 0 \end{bmatrix} \]

or

\[ \vec{B} = -3.81\hat{i} + 0\hat{j} + 0\hat{k} \]

or

\[ \vec{B} = \langle -3.81, 0, 0 \rangle \]

Vector Scaling can be calculated using the following equations:

\[ |k\vec{v}| = \sqrt{(k v_x)^2 + (k v_y)^2 + (k v_z)^2} \]

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

\[ |k\vec{v}| = \sqrt{k^2 (v_x^2 + v_y^2 + v_z^2)} \]

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

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

Now let's see what happens when we scale the red vector from the blue vector:

\[ \vec{A} = 3\hat{i} + 0\hat{j} + 0\hat{k} \]

\[ \vec{B} = -3.81\hat{i} + 0\hat{j} + 0\hat{k} \]

\[ |-1.27\vec{v}| = \sqrt{(-1.27 \cdot 3)^2 + (-1.27 \cdot 0)^2 + (-1.27 \cdot 0)^2} \]

\[ |-1.27\vec{v}| = \sqrt{-1.27^2 (3^2 +0^2 + 0^2)} \]

\[ |-1.27||\vec{v}| = \sqrt{-1.27^2} \sqrt{3^2 + 0^2 + 0^2} \]

\[ |-1.27||\vec{v}| = 1.27 \sqrt{3^2 + 0^2 + 0^2} \]

\[ |-1.27||\vec{v}| = 1.27 \sqrt{9 + 0 + 0} \]

\[ |-1.27||\vec{v}| = 1.27 \sqrt{9} \]

\[ |-1.27||\vec{v}| = 1.27 \cdot 3 \]

\[ |-1.27||\vec{v}| = 3.81 \]

Note: (The absolute value of -1.27 is 1.27)

\[ |\vec{v}| = 3 \]

As we can see from the formula, when we scale the red vector from the blue vector the magnitude is calculated to be 3, and when we scale the blue vector by a factor of 1.27 we obtain the red vector which has a magnitude of 3.81. Notice that the magnitude and scaled factor of the vector become positive, regardless of which direction a vector is pointing. This is important to understand, because although a "negative" vector is pointing in the opposite direction, or is scaled down, its magnitude is still a positive scalar. This is why a vector's direction is important to consider.

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