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.

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.

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 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 

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)

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 

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.

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 

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| )

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)

CROSS PRODUCT Length of the cross product

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)