The modulus of a vector, commonly known as its magnitude or length, quantifies how long the vector is. This measurement is a scalar quantity, which means it possesses only magnitude and lacks direction. The modulus of a vector is typically denoted as $|v|$ or $||v||$, where $v$ represents the vector itself.

To compute the modulus of a vector, you first need to identify its components. In two-dimensional space, a vector $v$ can be expressed as:

Here, $x$ and $y$ are the components of the vector along the x-axis and y-axis, respectively, while $i$ and $j$ are the unit vectors in those directions. In three-dimensional space, a vector $v$ is represented as:

In this case, $z$ is the component along the z-axis, and $k$ is the unit vector in that direction.

The formula for calculating the modulus of a vector in two-dimensional space is:

In three-dimensional space, the formula expands to:

These formulas are derived from the Pythagorean theorem, which asserts that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

For example, consider the vector $v = 3i + 4j$. The modulus of this vector can be calculated as follows:

Now, let’s examine another vector $v = 1i + 2j + 2k$. The modulus of this vector is computed as:

In summary, the modulus of a vector represents its length, calculated by taking the square root of the sum of the squares of its components. In two dimensions, this is expressed as $\sqrt{x^2 + y^2}$, while in three dimensions, it is represented as $\sqrt{x^2 + y^2 + z^2}$. This approach is consistent with the principles outlined in the Pythagorean theorem. For instance, for the vector $3i + 4j$ in two-dimensional space, the modulus is $5$, effectively illustrating how to determine the size of the vector.