Chapter 3 Euclidean Vector Spaces Vectors in nspace
- Slides: 21
Chapter 3 Euclidean Vector Spaces • Vectors in n-space • Norm, Dot Product, and Distance in n-space • Orthogonality • http: //www. traileraddict. com/clip/despicable-me/vectors-introduction
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 (a 1, a 2, …, an). 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, …, an) can be viewed either as a “generalized point” or as a “generalized vector”
Definition Two vectors u = (u 1 , u 2 , …, un) and v = (v 1 , v 2 , …, vn) in equal if u 1 = v 1 , u 2 = v 2 , …, un = vn are called The sum u + v is defined by u + v = (u 1+v 1 , …, un+vn) and if k is any scalar, the scalar multiple ku is defined by ku = (ku 1 , ku 2 , …, kun) Remarks The operations of addition and scalar multiplication in this definition are called the standard operations on.
The zero vector in 0 = (0, 0, …, 0). is denoted by 0 and is defined to be the vector If u = (u 1 , u 2 , …, un) 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 , …, -un). The difference of vectors in is defined by v – u = v + (-u) = (v 1 – u 1 , v 2 – u 2 , …, vn – un)
Theorem 3. 1. 1 (Properties of Vector in ) If u = (u 1 , u 2 , …, un), v = (v 1 , v 2 , …, vn), and w = (w 1 , w 2 , …, wn) 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) 1 u = u
Theorem 3. 1. 2 If v is a vector in , and k is a scalar, then a) 0 v = 0 b) k 0 = 0 c) (-1) v = - v Definition A vector w is a linear combination of the vectors v 1, v 2, …, vr if it can be expressed in the form w = k 1 v 1 + k 2 v 2 + · · · + kr vr where k 1, k 2, …, kr 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.
3. 2 Norm, Dot Product, and Distance in n-space Definition Example If u = (1, 3, -2, 7), then in the Euclidean space R 4 , the norm of u is
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.
Example: Find the unit vector u that has the same direction as v = (2, 2, -1). Solution: Definition, The standard unit vectors in Rn are: e 1 = (1, 0, … , 0), e 2 = (0, 1, …, 0), …, en = (0, 0, …, 1) In which case every vector v = (v 1, v 2, …, vn) in Rn can be expressed as v = (v 1, v 2, …, vn) = v 1 e 1 + v 2 e 2 +…+ vnen
Distance The distance between the points u = (u 1 , u 2 , …, un) and v = (v 1 , v 2 , …, vn) in Rn defined by Example If u = (1, 3, -2, 7) and v = (0, 7, 2, 2), then d(u, v) in R 4 is
Dot Product Example The dot product of the vectors u = (-1, 3, 5, 7) and v =(5, -4, 7, 0) in R 4 is
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 and 3. 2. 3 If u, v and w are vectors in a) b) c) d) e) f) g) h) i) and k is any scalar, then u · v = v · u u · (v+ w) = u · v + u · w k (u · v) = (ku) · v v · v ≥ 0; Further, v · v = 0 if and only if v = 0 0 · v = v · 0= 0 (u +v) · w = u · w + v · w u · (v- w) = u · v - u · w (u -v) · w = u · w - v · w k (u · v) = u · (kv) Example (3 u + 2 v) · (4 u + v) = (3 u) · (4 u + v) + (2 v) · (4 u + v ) = (3 u) · (4 u) + (3 u) · v + (2 v) · (4 u) + (2 v) · v =12(u · u) + 11(u · v) + 2(v · v)
Theorem 3. 2. 4 (Cauchy-Schwarz Inequality in ) If u = (u 1 , u 2 , …, un) and v = (v 1 , v 2 , …, vn) are vectors in |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) , then
Properties of Distance in If u, v, and w are vectors in and k is any scalar, then 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) Theorem 3. 2. 7 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: 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: .
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:
Lines and Planes Determined by Points and Normals Cont. Theorem 3. 3. 1 (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). Example: Determine whether the given planes are parallel. 4 x-y+2 z=5 and 7 x-3 y+4 z=8 Solution:
Orthogonal Projections Theorem 3. 3. 2 Projection Theorem If u and a are vectors in Rn, 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 projau, and 2. The vector w 2 is called the vector component of u orthogonal to a. Hence w 2=u-projau. In summary, (vector component of u along a) (vector component of u orthogonal to a)
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: Theorem 3. 3. 3 (Pythagorean Theorem in Rn) If u and v are orthogonal vectors in Rn with the Euclidean inner product, then
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