How do you determine the magnitude of a vector?

The magnitude of a vector is determined by calculating the square root of the sum of the squares of its components.

In more detail, a vector is a quantity that has both magnitude (size) and direction. The magnitude of a vector can be thought of as its 'length' in a certain direction. For example, if you have a vector that represents a displacement of 3 units east and 4 units north, the magnitude of this vector would represent the straight-line distance from the starting point to the ending point.

To calculate the magnitude of a vector, you need to know the components of the vector. These are the 'parts' of the vector in each direction. In a two-dimensional space, a vector has two components: one in the x-direction (horizontal) and one in the y-direction (vertical). In a three-dimensional space, a vector has three components: one in the x-direction, one in the y-direction, and one in the z-direction (depth).

The formula to calculate the magnitude of a vector in two dimensions is √(x² + y²), where x and y are the components of the vector. This comes 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 a vector in three dimensions, the formula is √(x² + y² + z²). This is an extension of the Pythagorean theorem to three dimensions.

In both cases, you square each component of the vector, add these squares together, and then take the square root of the result. This gives you the magnitude of the vector, which is always a positive number or zero. Remember, the magnitude of a vector gives you the 'length' of the vector, but not its direction. To fully describe a vector, you need both its magnitude and direction.