1. 2005/8Vectors-1 Vectors in n-space

2. 2005/8Vectors-2 Definition of points in space A pair of numbere (x, y) can be used to represent a point in the plane. Define a point X in n-space to be an n-tuple of numbers (x 1, x 2, …, x n ). The number x i is called the coordinates( 座標 ) of the point X. We denote n-space by R n. x y  (x, y)

3. 2005/8Vectors-3 Most of our examples will take place when n=2 or n=3. The case n=4 does occur in physics. Don’t get the idea that “time is the fourth dimension”. The above 4-dimensional space is only one possible example. We can view a higher dimensional space as “products” of lower dimensional space. R 3 = R 2  R 1 ; R 4 = R 3  R 1 ; R 4 = R 2  R 2

4. 2005/8Vectors-4 The operations of the points with coordinates Assume that A=(a 1, a 2, …, a n ) and B=(b 1, b 2, …, b n ) are two points in n-space. Define A+B = (a 1 + b 1, a 2 + b 2, …, a n + b n ). Example If A = (1, 2) and B = (  3, 5), then A+B = (1+(  3), 2+5) = (  2, 7).

5. 2005/8Vectors-5 The Properties of Addition The following rules are satisfied: 1. (A + B) + C = A + (B + C) 2. A + B = B + A 3. Let 0 = (0, 0, …,0). Then 0 + A = A for all A. 4. Let A=(a 1, a 2, …, a n ) and  A=(  a 1,  a 2, …,  a n ). Then A + (  A) = 0. Pf: Omit.

6. 2005/8Vectors-6 View Addition Geometrically (  1, 1)   (1, 4)  (2, 3)  A A BB  A+B The figure looks like a parallelogram.

7. 2005/8Vectors-7 Multiplication by a Number If c is any number (or scalar), and A=(a 1, a 2, …, a n ) is a point. We define cA = (ca 1, ca 2, …, ca n ) Example If A = (2,  1, 5) and c = 7, then cA = (7  2, 7  (  1), 7  5) = (14,  7, 35).

8. 2005/8Vectors-8 The Properties of Multiplication It is easy to verify the rules: 5. c(A+B) = cA + cB. 6. If c 1, c 2 are two numbers, then (c 1 +c 2 )A = c 1 A + c 2 A and (c 1 c 2 )A = c 1 (c 2 A) Pf: Omit. Note: (  1)A =  A.

9. 2005/8Vectors-9 View Multiplication Geometrically  A = (1, 2)  2A = (2, 4)  ⅓ A = (⅓, ⅔)   ⅓ A = (  ⅓,  ⅔) A = cB: c >0, they have the same direction from origin. c <0, they have the opposite direction from origin.

10. 2005/8Vectors-10 Located Vectors Defined a located vector, AB = B + (  A), A is called the beginning point and B is called the end point. Example: P = (1,  1, 3), Q = (2, 4, 1). Then PQ = (1  2,  1  4, 3  1) = (  1,  5, 2). AB and CD are equivalent if B  A = D  C. Every located vector is equivalent to one vector whose beginning point is the origin. who is located at the origin.

11. 2005/8Vectors-11 Let P = (1,  1, 3) and Q =(2, 4, 1). Then PQ is equivalent to OC, where C = Q  P = (1, 5,  2). If A = (4,  2, 5) and B = (5, 3, 3), then PQ is equivalent to AB because B  A = (1, 5,  2) = Q  P. Two located vectors AB and PQ are said to be parallel if  c  0 such that B  A = c(Q  P). Same direction if c>0, opposite direction if c<0.