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Section 3.2

Section 3.2. Norm of a Vector; Vector Arithmetic. PROPERTIES OF VECTOR ARITHMETIC. Theorem 3.2.1 : If u , v , and w are vectors in 2- or 3-space and k and l are scalars, then the following relationships hold. (a) u + v = v + u (b) ( u + v ) + w = u + ( v + w )

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Section 3.2

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  1. Section 3.2 Norm of a Vector; Vector Arithmetic

  2. PROPERTIES OF VECTOR ARITHMETIC Theorem 3.2.1: If u, v, and w are vectors in 2- or 3-space and k and l are scalars, then the following relationships hold. (a) u + v = v + u (b) (u + v) + w = u + (v + w) (c) u + 0 = 0 + u = u (d) u + (−u) = 0 (e) k(lu) = (kl)u (f) k(u + v) = ku + kv (g) (k + l)u = ku + lu (h) 1u = u

  3. NORM OF A VECTOR The length of a vector u is often called the norm of u and is denoted by ||u||. By the Pythagorean Theorem, we have

  4. DISTANCE BETWEEN POINTS If P1(x1, y1, z1) and P2(x2, y2, z2) are two points in 3-space, then the distance between them is the norm of the vector . That is, A similar result holds for the distance between two points in 2-space.

  5. REMARKS ABOUT THE NORM • A vector that has norm 1 is called a unit vector. • From the definition of ku and the definition of norm, we have ||ku|| = |k| ||u||

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