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Vector Scaling

Vector Scaling in 3D

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 the same simulation as the ne previously of vector scaling in 3D. There is a red vector of magnitude (length) 2.915, and a second vector, a blue vector of magnitude (length) 5.831. In the perspective of the red vector, it is half of the magnitude (length) of the blue vector. In the perspective of the blue vector, it is double the magnitude (length) of the red vector. These vectors maintain the same angle and direction. This simulation allows you as the student to change the scale factor, and thus the magnitude of the vector.

Red Vector:

x, y, z:

Scale:

Scale:

Blue Vector:

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

\[ \vec{A} = \begin{bmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1.5 & 1.5 & 2 \end{bmatrix} \]

or

\[ \vec{A} = 1.5\hat{i} + 1.5\hat{j} + 2\hat{k} \]

or

\[ \vec{A} = \langle 1.5, 1.5, 2 \rangle \]

and

\[ \vec{B} = \begin{bmatrix} \hat{i} & \hat{j} & \hat{k} \\ 3 & 3 & 4 \end{bmatrix} \]

or

\[ \vec{B} = 3\hat{i} + 3\hat{j} + 4\hat{k} \]

or

\[ \vec{B} = \langle 3, 3, 4 \rangle \]

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:

\[ \vec{A} = 1.5\hat{i} + 1.5\hat{j} + 2\hat{k} \]

\[ \vec{B} = 3\hat{i} + 3\hat{j} + 4\hat{k} \]

\[ |2\vec{v}| = \sqrt{(2 \cdot 1.5)^2 + (2 \cdot 1.5)^2 + (2 \cdot 2)^2} \]

\[ |2\vec{v}| = \sqrt{2^2 (1.5^2 + 1.5^2 + 2^2)} \]

\[ |2||\vec{v}| = \sqrt{2^2} \sqrt{1.5^2 + 1.5^2 + 2^2} \]

\[ |2||\vec{v}| = 2 \sqrt{1.5^2 + 1.5^2 + 2^2} \]

\[ |2||\vec{v}| = 2 \sqrt{2.25 + 2.25 + 4} \]

\[ |2||\vec{v}| = 2 \sqrt{8.5} \]

\[ |2||\vec{v}| = 2 \cdot 2.915 \]

\[ |2||\vec{v}| = 5.831 \]

\[ |\vec{v}| = 2.915 \]

As we can see from the formula, when we scale the red vector from the blue vector the magnitude is calculated to be 2.915, and when we scale the blue vector by a factor of 2 we obtain the red vector which has a magnitude of 5.831.

You have now learned what vector scaling is. Go back now to the vector page to move onto the next lesson.

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