vectors (ch. 12) vectors in the plane definition: a vector v in the cartesian plane is an ordered...
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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.OP
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 = v1, v2 and u = u1, u2 are equal if v1= u1 and v2 = u2.
2. The sum of two vectors v = v1, v2 and u = u1, u2: v + u = v1 + u1 , v2 + u2
3. If u = u1, u2 and c is a real number, then the scalar multiple cu is the vector
cu = cu1, cu2
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 = a1, a2 0 then
- Special unit vectors: i = 1, 0 and j = 0, 1
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If a = a1, a2 , then
a = a1, 0 + 0, a2 = a1 1, 0 + a2 0, 1
= a1 i + a2 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
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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)
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) = (ac)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)