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# Calculating the Length of a Vector

The length of a vector v is often called the v norm and is expressed as |v|. Let v = (v1, v2) be a vector in space 2, then the norm of vector v is expressed as,
And it is illustrated with Figure 1,

Suppose that v = (v1, v2, v3) is a vector in space 3. Using Figure 2,
Then we get it

so,

If P1(x1, y1, z1) and P2(x2, y2, z2) are two points in space 3, then the distance d between the two points is the length of the vector P1P2 (Figure 3),

because,

So based on the norm vector formula in Space 3, it is clear that

Likewise P1 (x1, y1) and P2 (y1, y2) are two points in space 2, then the distance between the two points is determined by:

Example 1.

The length of the vector v = (-3, 2, 1) is

The distance d between points P1 (2, -1, -5) and point P2 (4, -3, 1) are