Chapter 3 Euclidean Vector Spaces Vectors in n-space Norm, Dot Product, and Distance in n-space Orthogonality

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

Definition Two vectors u = (u 1,u 2,…,u n ) and v = (v 1,v 2,…, v n ) in are called equal if u 1 = v 1,u 2 = v 2, …, u n = v n The sum u + v is defined by u + v = (u 1 +v 1, u 1 +v 1, …, u n +v n ) and if k is any scalar, the scalar multiple ku is defined by ku = (ku 1,ku 2,…,ku n ) Remarks The operations of addition and scalar multiplication in this definition are called the standard operations on.

If u = (u 1,u 2,…,u n ) is any vector in, then the negative (or additive inverse) of u is denoted by -u and is defined by -u = (-u 1,-u 2,…,-u n ). The zero vector in is denoted by 0 and is defined to be the vector 0 = (0, 0, …, 0). The difference of vectors in is defined by v – u = v + (-u) = (v 1 – u 1,v 2 – u 2,…,v n – u n )

Theorem (Properties of Vector in ) If u = (u 1,u 2,…,u n ), v = (v 1,v 2,…, v n ), and w = (w 1,w 2,…,w n ) are vectors in and k and m are scalars, then: a) u + v = v + u b) u + (v + w) = (u + v) + w c) u + 0 = 0 + u = u d) u + (-u) = 0; that is, u – u = 0 e) k(mu) = (km)u f) k(u + v) = ku + kv g) (k+m)u = ku+mu h) 1u = u

Theorem If v is a vector in, and k is a scalar, then a) 0v = 0 b) k0 = 0 c) (-1) v = - v Definition A vector w is a linear combination of the vectors v 1, v 2,…, v r if it can be expressed in the form w = k 1 v 1 + k 2 v 2 + · · · + k r v r where k 1, k 2, …, k r are 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.

Definition 3.2 Norm, Dot Product, and Distance in n-space Example If u = (1,3,-2,7), then in the Euclidean space R 4, the norm of u is

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

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 R n are: e 1 = (1, 0, …, 0), e 2 = (0, 1, …, 0), …, e n = (0, 0, …, 1) In which case every vector v = (v 1,v 2, …, v n ) in R n can be expressed as v = (v 1,v 2, …, v n ) = v 1 e 1 + v 2 e 2 +…+ v n e n

The distance between the points u = (u 1,u 2,…,u n ) and v = (v 1, v 2,…,v n ) in R n defined by Example If u = (1,3,-2,7) and v = (0,7,2,2), then d(u, v) in R 4 is Distance

Dot Product Example The dot product of the vectors u = (-1,3,5,7) and v =(5,-4,7,0) in R 4 is u · v = (-1)(5) + (3)(-4) + (5)(7) + (7)(0) = 18

It is common to refer to, with the operations of addition, scalar multiplication, and the Euclidean inner product, as Euclidean n-space. Theorem and If u, v and w are vectors in and k is any scalar, then a) u · v = v · u b) u · (v+ w) = u · v + u · w c) k (u · v) = (ku) · v d) v · v ≥ 0; Further, v · v = 0 if and only if v = 0 e) 0 · v = v · 0= 0 f)(u +v) · w = u · w + v · w g) u · (v- w) = u · v - u · w h)(u -v) · w = u · w - v · w i) k (u · v) = u · (kv) 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)

Theorem (Cauchy-Schwarz Inequality in ) If u = (u 1,u 2,…,u n ) and v = (v 1, v 2,…,v n ) 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 a) || u || ≥ 0 b) || u || = 0 if and only if u = 0 c) || ku || = | k ||| u || d) || u + v || ≤ || u || + || v || (Triangle inequality for vectors)

a) d(u, v) ≥ 0 b) d(u, v) = 0 if and only if u = v c) d(u, v) = d(v, u) d) d(u, v) ≤ d(u, w ) + d(w, v) (Triangle inequality for distances) Properties of Distance in If u, v, and w are vectors in and k is any scalar, then Theorem If u, v, and w are vectors in with the Euclidean inner product, then

Dot Products as Matrix Multiplication

3.3 Orthogonality Example In the Euclidean space, determine if the vectors u = (-2, 3, 1, 4) and v = (1, 2, 0, -1) are orthogonal. Solution: since u · v = (-2)(1) + (3)(2) + (1)(0) + (4)(-1) = 0, u and v are orthogonal. Example In the Euclidean space R 3, determine if the standard unit vectors i=(1, 0, 0), j=(0, 1, 0), k=(0, 0, 1) is an orthogonal set. Solution: we must show that i · j = i ·k = j ·k = 0.

Lines and Planes Determined by Points and Normals A line in R 2 is determined uniquely by its slope and one of its points, and that a plane in R 3 is determined uniquely by its “inclination” and one of its points. One way of specifying slope and inclination is to use a nonzero vector n, called normal, that is orthogonal to the line or plane in question. The point-normal equation of the line through the point P 0 (x 0, y 0 ) that has normal n=(a, b) is: a(x-x 0 )+b(y-y 0 )=0 The point-normal equation of the plane through the point P 0 (x 0, y 0, z 0 ) that has normal n=(a, b, c) is a(x-x 0 )+b(y-y 0 )+c(z-z 0 )=0 Example Find a point-normal equation of the plane through the point P(-1, 3, -2) that has normal n=(-2, 1, -1). Solution:

Theorem (a)If a and b are constants that are not both zero, then an equation of the form ax+by+c=0 represents a line in R 2 with normal n=(a, b). (b) If a, b, and c are constant that are not all zero, then an equation of the form ax+by+cz+d=0 represents a plane in R 3 with normal n=(a, b, c). Lines and Planes Determined by Points and Normals Cont. Example: Determine whether the given planes are parallel. 4x-y+2z=5 and 7x-3y+4z=8 Solution:

Orthogonal Projections In summary, (vector component of u along a) (vector component of u orthogonal to a) Theorem Projection Theorem If u and a are vectors in R n, and if a  o, then u can be expressed in exactly one way in the form u=w 1 +w 2, where w 1 is a scalar multiple of a and w 2 is orthogonal to a. Note: 1.Here the vector w 1 is called the orthogonal projection of u on a, or sometimes the vector component of u along a, denoted by proj a u, and 2.The vector w 2 is called the vector component of u orthogonal to a. Hence w 2 =u-proj a u.

Theorem (Pythagorean Theorem in R n ) If u and v are orthogonal vectors in R n with the Euclidean inner product, then Example Let u=(2, -1, 3) and a=(4, -1, 2). Find the vector component of u along a and the vector component of u orthogonal to a. Solution: