### Analytic Geometry

#### Vector Length

##### Pythagorean Theorem

The Pythagorean theorem states that for any right triangle, the sum of the squares of the shorter sides (which are adjacent to the right triangle) is equal to the square of the diagonal side (which is opposite the right angle and is also referred to as the hypotenuse or radius):

Figure 1: Pythagorean theorem.

Since the shorter sides of a right triangle are perpendicular, they can form horizontal and vertical offsets for a point on a circle, with the hypotenuse as the radius of the circle:

Figure 2: Circle.

Hence, the Pythagorean theorem forms the basis of the equation of a circle.

The equation of a circle with center at the origin is:

x² + y² = r²

The equation of a circle with center at (a, b) is:

(x − a)² + (y − b)² = r²

which becomes the equation of a circle with center at the origin when a = 0 and b = 0.

The Pythagorean theorem extends to more dimensions. In three dimensions:

r²  =  x² + y² + z²

where x, y and z are perpendicular distances, for example width, height and depth. This forms the basis of the equation of a sphere.

##### Vector Length

The length of a vector is the distance from the tail to the head of the vector. That distance is calculated using the Pythagorean theorem, by summing the squares of the components of the vector, and taking the principal square root of that sum.

We illustrate with the vector (3, 4):

Figure 3:  Vector v = (3, 4).

Using the Pythagorean theorem to calculate the length of the vector v = (3, 4):

|𝙫|
=  $\sqrt{3·3 + 4·4}$
=  $\sqrt{9 + 16}$
=  $\sqrt{25}$
=  5

Vertical bars denote the length of a vector, like absolute value on the number line (see Figure 9 o in the Number Line lesson).

Indeed, absolute value on a number line is the Pythagorean theorem in one dimension (length of a vector with one component). Mathematically, the absolute value of a number in one dimension is the principal square root of the square of the number.

##### Unit Vector

The length of a vector is also called the vector norm. A vector with length (norm) equal to 1 is called a unit vector, or normalized vector.

Converting a vector to a unit vector is called normalizing the vector. A normalized vector, which may be denoted with a hat ( ^ ), is the same direction as the original vector, but has a vector length of one regardless of what the vector length of the original vector was.

To normalize a vector, simply divide each component by the vector length, which is scalar multiplication of the vector with the reciprocal of the vector length.

For example, to normalize u = (x, y):

###### Analytic Geometry
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Number Line
Geometric Vectors
Angles
Squares & Cubes