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VECTORS (Ch. 12) Vectors in the plane Definition: A vector v in the Cartesian plane is an ordered pair of real numbers: a,b . We write v = a,b and call a and b the components of the vector v. Geometric representation of vector v is the directed line segment from origin O to point P( a,b ). A directed line segment has length and direction.

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Length of a vector v = a,b is defined as Example: The length of the vector v = 2,-3 is The zero vector is 0= 0,0 has length zero and no direction.

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Algebraic operations 1. Two vectors v = v 1, v 2 and u = u 1, u 2 are equal if v 1 = u 1 and v 2 = u 2. 2. The sum of two vectors v = v 1, v 2 and u = u 1, u 2 : v + u = v 1 + u 1, v 2 + u 2 3. If u = u 1, u 2 and c is a real number, then the scalar multiple c u is the vector c u = cu 1, cu 2

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Let a, b and c be vectors and r and s real numbers. Then 1. a + b = b + a 2. a + (b + c) = (a + b) + c 3. r(a + b) = r b + r a 4. (r + s) a = r a + s a 5. (rs) a = r(sa) = s(ra)

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The unit vectors i and j – A unit vector is a vector of length 1. – If a = a 1, a 2 0 then – Special unit vectors: i = 1, 0 and j = 0, 1

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If a = a 1, a 2 , then a = a 1, 0 + 0, a 2 = a 1 1, 0 + a 2 0, 1 = a 1 i + a 2 j. So every vector in the plane is a linear combination of i and j.

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Vectors in space (3- dimensional) v = x, y, z with length: can be written: v = x i + y j + z k, where i = 1, 0, 0 , j = 0, 1, 0 , k = 0, 0, 1

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Dot Product Given two vectors Theorem. If is the angle between a and b then the dot product is defined as a b = |a| |b| cos . cos = ( a b) / ( |a| |b| )

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The angle between a and b can be found using cos = ( a b) / ( |a| |b| ). Properties: Let a and b be vectors and r a real number. Then a b = b a a a = | a| 2 a ( b + c ) = a b + a c ( r a ) b = r (a b ) = a (r b)

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CROSS PRODUCT Length of the cross product

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Properties a, b and c are vectors and r is a real number. 1. a x b = -(b x a) 2. a x (b x c) = (a c)b – (a b)c 3. a (b x c) = (a x b) c 4. a x ( b + c ) = (a x b) + (a x c) 5. ( r a ) x b = r (a x b ) = a x (r b)

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