How do you find the modulus of a vector?

The modulus of a vector is found by calculating the square root of the sum of the squares of its components.

In more detail, the modulus of a vector, often referred to as the magnitude or length of a vector, is a measure of how long the vector is. It is a scalar quantity, meaning it only has magnitude and no direction. The modulus of a vector is denoted by |v| or ||v||, where v is the vector.

To calculate the modulus of a vector, you need to know the components of the vector. In a two-dimensional space, a vector v is usually represented as v = xi + yj, where x and y are the components of the vector in the x and y directions respectively, and i and j are the unit vectors in the x and y directions. In a three-dimensional space, a vector v is represented as v = xi + yj + zk, where z is the component of the vector in the z direction and k is the unit vector in the z direction.

The formula to calculate the modulus of a vector in a two-dimensional space is |v| = √(x² + y²), and in a three-dimensional space, the formula is |v| = √(x² + y² + z²). This is derived from the Pythagorean theorem, which states that in a right-angled triangle, the square of the length 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, if you have a vector v = 3i + 4j, the modulus of the vector is |v| = √(3² + 4²) = √(9 + 16) = √25 = 5. If you have a vector v = 1i + 2j + 2k, the modulus of the vector is |v| = √(1² + 2² + 2²) = √(1 + 4 + 4) = √9 = 3.

IB Physics Tutor Summary: The modulus of a vector is its length, calculated by taking the square root of the sum of its components squared. In 2D, it's √(x² + y²) and in 3D, √(x² + y² + z²), similar to the Pythagorean theorem. For instance, a vector 3i + 4j in 2D has a modulus of 5, showing how you can determine the vector's size.