VECTORS IN TWO DIMENSIONS

scalar quantity	scalar	directed line segment
initial point	terminal point	magnitude
equivalent vectors	equal vectors	velocity vector
force vector	displacement	sum of vectors
resultant force	scalar multiple	position vector
components of vector	a = ( a1,a2 )

Notation: is the set of all vectors ( x,y ) where x and y are real numbers.

Definition: The magnitude ||a|| of the vector a = ( a1,a2 ) is ||a|| =

Definition: (Addition of Vectors): ( a1,a2 ) + ( b1,b2 ) = ( a1 + b1, a2 + b2 )

Definition: (Scalar Multiple of a Vector): c ( a1,a2 ) = ( ca1,ca2 )

Definition: The zero vector 0 is O = ( 0,0 ),

and the negative of a is - a = - ( a1,a2 ) = ( -a1, -a2 )

Theorem: The follow properties hold for vectors a, b, w and all scalars c and d:

a + b = b + a	a + (b+w) = (a + b) + w	a + 0 = a
a + (-a) = 0	c (a+b) = ca + cb	(c+d)a = ca + da
(cd)a = c(da) = d(ca)	1a = a	0a = 0 = c0

Definition: If a = (a1,a2) and b = (b1,b2), the difference a - b is a + (-b).

Theorem: If P1(x1,y1) and P2(x2,y2) are any two points, then the vector a in that corresponds to P1P2 is a = ( x2 - x1, y2 - y1 ).

Definition: Nonzero vectors a and b in have

(i) the same direction if b = ca for some positive scalar c > 0.

(ii) the opposite direction if b = ca for some negative scalar c < 0.

The vectors a and b are parallel if b = ca for some scalar c

Theorem: If a is a vector and c is a scalar, then || c a || = |c| || a ||

Definition: i = ( 1,0 ), j = ( 0,1 )

Theorem: If a = ( a1,a2 ), then a = a1i + a2j

Definition: If a = ( a1,a2 ), then a1 is the horizontal component of a and a2 is the vertical component of a.

VECTORS IN THREE DIMENSIONS

ordered triple	xy-plane	x- coordinate
equality of triples	yz-plane	y- coordinate
origin	xz-plane	z-coordinate
right-handed coordinate system	rectangular coordinate system	xyz-coordinate system

Distance Formula: The distance between and is

Midpoint Formula: The midpoint of the line segment from to is

Definition: The graph of an equation in three variables x, y, and z is the set of all points P(a,b,c) in a rectangular coordinate system such that the ordered triple (a,b,c) is a solution of the equation. The graph of such an equation is called a surface.

Theorem: An equation of a sphere of radius r and center is

Definition: is the set of all vectors (x,y,z ) where x, y and z are real numbers.

Definition: In, i = ( 1,0,0 ), j = ( 0,1,0 ), and k = ( 0,0, 1).

Note: We may write the vector a = ( a1, a2, a3 ) as a = a1i + a2j + a3k