2433-Ch12-slides - 3-D Coordinate System(12.1 P(x1 y1 z1 and Q(x2 y2 z2 points in 3-space(A Distance Formula d(P Q =(x2 x1)2(y2 y1)2(z2 z1)2(B Midpoint

# 2433-Ch12-slides - 3-D Coordinate System(12.1 P(x1 y1 z1...

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3-D Coordinate System: (12.1) P : ( x 1 , y 1 , z 1 ) and Q : ( x 2 , y 2 , z 2 ), points in 3-space: (A) Distance Formula: d ( P, Q ) = ( x 2 - x 1 ) 2 + ( y 2 - y 1 ) 2 + ( z 2 - z 1 ) 2 (B) Midpoint Formula: The mid- point R of the line segment joining P and Q is the point R : x 1 + x 2 2 , y 1 + y 2 2 , z 1 + z 2 2 . 1
Examples: 1.The pointsA: (2,-1,3)andB: (-4,5,-1)are the endpoints of a diameter ofa sphere.Find an equation for thesphere. 2
2.The pointsA: (2,-2,1), B: (1,1,3), C: (2,0,5)are the vertices of a right triangle.Find an equation of the sphere withcenter at the midpoint of the hy-potenuse and passing through thevertex opposite the hypotenuse.
Vectors: (12.2, 12.3) A vector in n -dimensional space is a directed line segment; it is repre- sented by an ordered n -tuple of real numbers. 3-Space: A vector a in 3-space is an ordered triple of numbers: a = ( a 1 , a 2 , a 3 ) The vector 0 = (0 , 0 , 0) is the zero vector . 3
Operations on Vectors Let a = ( a 1 , a 2 , a 3 ) and b = ( b 1 , b 2 , b 3 ) be vectors in 3-space and let α be a real number (scalar). (A) Equality: a = b iff a 1 = b 1 , a 2 = b 2 , a 3 = b 3 . 4
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Properties of Addition: Let a , b , c be vectors. 1. a + b = b + a (commutative) 2. ( a + b ) + c = a + ( b + c ) (associative) 3. a + 0 = 0 + a = a , 0 is the additive identity 6
4.to each vectorathere corre-sponds a uniquexsuch thata+x=x+a=0(additive inverse)xis denoted-a
Subtraction: a - b = a + ( - b ) = ( a 1 - b 1 , a 2 - b 2 , a 3 - b 3 ) 7
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3.(αβ)a=α(βa) =β(αa) NOTES: 1. a and b are parallel iff a = λ b for some number λ . 2. 0 is parallel to every vector; 0 = 0 a for all a .
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Properties of Norm: 1. a 0; a = 0 iff a = 0 . 2. α a = | α | a . 3. a + b a + b . (triangle inequality) 10
Unit Vectors: 11
Unit Coordinate Vectors: i = (1 , 0 , 0) , j = (0 , 1 , 0) , k = (0 , 0 , 1) i , j , k -Representation: a = ( a 1 , a 2 , a 3 ) = a 1 i + a 2 j + a 3 k Direction: ???? 0 has no direction. 12
Dot Product: (12.4) Let a = ( a 1 , a 2 , a 3 ) and b = ( b 1 , b 2 , b 3 ) be vectors. The
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