Understanding Vectors in n-Space: Norm, Dot Product, and Orthogonality

1. Chapter 3 • Vectors in n-space • Norm, Dot Product, and Distance in n-space • Orthogonality

2. 3. 1 Vectors in n-space Definition If n is a positive integer, then an ordered n-tuple is a sequence of n real numbers (a1,a2,…,an). The set of all ordered n-tuple is called n-space and is denoted by . Note that an ordered n-tuple (a1,a2,…,an) can be viewed either as a “generalized point” or as a “generalized vector”

3. Definition Two vectors u = (u1,u2,…,un) and v = (v1,v2,…, vn) in are called equal if u1 = v1,u2 = v2, …, un = vn The sum u + v is defined by u + v = (u1+v1, u1+v1, …, un+vn) and if k is any scalar, the scalar multiple ku is defined by ku = (ku1,ku2,…,kun) Remarks The operations of addition and scalar multiplication in this definition are called the standard operations on .

4. The zero vector in is denoted by 0 and is defined to be the vector 0 = (0, 0, …, 0). If u = (u1,u2,…,un) is any vector in , then the negative (or additive inverse) of u is denoted by -u and is defined by -u = (-u1,-u2,…,-un). The difference of vectors in is defined by v – u = v + (-u) = (v1 – u1,v2 – u2,…,vn– un)

5. Theorem 3. 1.1 (Properties of Vector in ) If u = (u1,u2,…,un), v = (v1,v2,…, vn), and w = (w1,w2,…,wn) are vectors in and k and m are scalars, then: • u + v = v + u • u + (v + w) = (u + v) + w • u + 0 = 0 + u = u • u + (-u) = 0; that is, u – u = 0 • k(mu) = (km)u • k(u + v) = ku + kv • (k+m)u = ku+mu • 1u = u

6. Theorem 3. 1.2 If v is a vector in , and k is a scalar, then • 0v = 0 • k0 = 0+ (v + w) = (u + v) + w • (-1) v = - v Definition A vector w is a linear combination of the vectors v1, v2,…, vrif it can be expressed in the form w = k1v1 + k2v2 + · · · + kr vr where k1, k2, …, krare scalars. These scalars are called the coefficients of the linear combination. Note that the linear combination of a single vector is just a scalar multiple of that vector.

7. 3.2 Norm, Dot Product, and Distance in n-space Definition Example If u = (1,3,-2,7), then in the Euclidean space R4 , the norm of u is

8. Normalizing a Vector Definition A vector of norm 1 is called a unit vector. That is, if v is any nonzero vector in Rn , then The process of multiplying a nonzero vector by the reciprocal of its length to obtain a unit vector is called normalizing v.

9. Example: Find the unit vector u that has the same direction as v = (2, 2, -1). Solution: The vector v has length Thus, Definition, The standard unit vectors in Rn are: e1 = (1, 0, … , 0), e2 = (0, 1, …, 0), …, en = (0, 0, …, 1) In which case every vector v = (v1,v2, …, vn) in Rn can be expressed as v = (v1,v2, …, vn) = v1e1 + v2e2 +…+ vnen

10. Distance The distance between the points u = (u1,u2,…,un) and v = (v1, v2,…,vn) in Rn defined by Example If u = (1,3,-2,7) and v = (0,7,2,2), then d(u, v) in R4 is

11. Dot Product Definition If u = (u1,u2,…,un), v = (v1,v2,…, vn) are vectors in , then the dot product u · v is defined by u · v = u1v1+ u2v2+… + un vn Example The dot product of the vectors u = (-1,3,5,7) and v =(5,-4,7,0) in R4 is u · v = (-1)(5) + (3)(-4) + (5)(7) + (7)(0) = 18

12. It is common to refer to , with the operations of addition, scalar multiplication, and the Euclidean inner product, as Euclidean n-space. Theorem 3.2.2 If u, v and w are vectors in and k is any scalar, then • u · v = v · u • (u + v) · w = u · w + v · w • (k u) · v = k(u · v) • v · v ≥ 0; Further, v · v = 0 if and only if v = 0 Example (3u + 2v) · (4u + v) = (3u) · (4u + v) + (2v) · (4u + v ) = (3u) · (4u) + (3u) · v + (2v) · (4u) + (2v) · v =12(u · u) + 11(u · v) + 2(v · v)

13. Theorem 3.2.4 (Cauchy-Schwarz Inequality in ) If u = (u1,u2,…,un) and v = (v1, v2,…,vn) are vectors in , then |u · v| ≤ || u || || v || Or in terms of components Properties of Length in ) If u and v are vectors in and k is any scalar, then • || u || ≥ 0 • || u || = 0 if and only if u = 0 • || ku || = | k ||| u || • || u + v || ≤ || u || + || v || (Triangle inequality)

14. Properties of Distance in If u, v, and w are vectors in and k is any scalar, then • d(u, v) ≥ 0 • d(u, v) = 0 if and only if u = v • d(u, v) = d(v, u) • d(u, v) ≤ d(u, w ) + d(w, v) (Triangle inequality) Theorem 3.2.7 If u, v, and w are vectors in with the Euclidean inner product, then

15. Dot Products as Matrix Multiplication

16. 3.3 Orthogonality Example In the Euclidean space the vectors u = (-2, 3, 1, 4) and v = (1, 2, 0, -1) are orthogonal, since u · v = (-2)(1) + (3)(2) + (1)(0) + (4)(-1) = 0 Theorem 3.3.3 (Pythagorean Theorem in ) If u and v are orthogonal vectors in with the Euclidean inner product, then