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Vector Magnitude

Vector Magnitude in 3D

The magnitude of a vector is the numerical value length that a vector possesses. It is always a positive scalar regardless of direction or angle.

Below is the same simulation as the one previously of a vector in the x, y, and z axis. It has a magnitude (length) of 5.831, an angle of 43.31, and its direction is northeast. As a student you will be able to change this vector's angle, direction, and magnitude. Regardless of which direction the 3D vector will be pointing, whether that be positive or negative, or any angle it is in, the magnitude will always be a constant scalar.

Red Vector:

x, y, z:

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

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

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

\[ |\vec{v}| = \sqrt{3^2 + 3^2 + 4^2} \]

\[ |\vec{v}| = \sqrt{9 + 9 + 16} \]

\[ |\vec{v}| = \sqrt{34} \]

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

As we can see, the magnitude (length) is 5.831, just like in the simulation. Note that the x-axis vector component is 3, the y-axis vector component is 3, and the z-axis vector component is 4. This is because the vector is on the x, y, and z axis.

You have now learned about a vector's magnitude. Go back now to the vector page to move onto the next lesson.

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