Length and Dot Product in R^n

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Length and Direction

Vectors can be characterized by two geometric quantities: length and direction. We begin by reviewing the definition of the length of a vector in R2.

Let v = (v1,v2) be a vector in R2.

If the vector is represented geometrically with its initial point at the origin, then its length is the length of the hypotenuse of a right triangle whose legs have lengths |v1| and |v2|.

<p>Vectors can be characterized by two geometric quantities: length and direction. We begin by reviewing the definition of the length of a vector in R<sup>2</sup>. </p><p>Let v = (v<sub>1</sub>,v<sub>2</sub>) be a vector in R<sup>2</sup>. </p><p>If the vector is represented geometrically with its initial point at the origin, then its length is the length of the hypotenuse of a right triangle whose legs have lengths |v<sub>1</sub>| and |v<sub>2</sub>|.</p>
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Length / Norm / Magnitude

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Unit Vector

If ||v|| =1, then v is called a unit vector.

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Non-negativity (||v|| ≥ 0)

Because the square of each component is nonnegative, ||v|| ≥ 0 for every vector v.

Moreover, ||v|| ≥ 0 if and only if v1 = v2 = … = vn = 0.

Therefore, ||v|| = 0 ⇐⇒ v = 0.