7/28/2019 Analytic Geometry Summary Sheet
1/2
Mr. Simonds MTH 253
V e c t o r B a s i c s s t u f f t o b e m e m o r i z e d | 1
Basic Vector Language and Vector Arithmetic
Fact 1: The three standard unit vectors are 1,0,0i = , 0,1,0j = and 0,0,1k = . The
vector1 2 3, ,u u u u=
G
can also be written as 1 2 3 u u i u j u k = + +
G
. Consequently, the
components of the vector u
G
; 1u , 2u , and 3u ; are called, respectively, the i component,the j component, and the k component.
Fact 2: To add or subtract 2 vectors you add or subtract the corresponding components of the
two vectors. To multiply a vector by a scalar (number), you multiply each component of
the vector by the scalar.
Fact 3: Two non-zero vectors are called parallel if and only if they point in exactly the same
direction or they point in exactly opposite directions.
Fact 4: A non-zero vector is said to be parallel to a line if and only if the vector couldbe drawn
entirely along the line.
Fact 5: A non-zero vector is said to be parallel to a plane if and only if the vector couldbe drawn
entirely on the plane.
Fact 6: A non-zero vector is said to be perpendicular (or normal) to a plane if and only if the
vector forms a right angle when drawn tail-to-tail with any vector parallel to the plane.
Vector Norms
Def 1: The norm of the vector 1 2 3, ,u u u u=G
is 2 2 21 2 3u u u u= + +G
. In 2-dimensions and 3-
dimenstions this quantity is commonly referred to as the length of the vector. In appliedproblems this quantity is commonly referred to as the magnitude of the vector.
Fact 7: A vector, vG
, is called a unit vector if and only if 1v =G
. Unit vectors are generally
annotated thusly: v .
Fact 8: For a non-zero vector vG
, v
uv
=
G
Gis the unit vector that points in the same direction as
vG
. The process of dividing a vector by its length is called normalizing the vector.
The Dot Product
Def 2: If 1 2 3, ,u u u u=G
and 1 2 3, ,v v v v=G
, then 1 1 2 2 3 3u v u v u v u v = + +G G
.
Fact 9: The smallest angle, , formed by non-zero vectors uG
and vG
when uG
and vG
are drawn
tail-to-tail satisfies the equation ( )cosu v
u v
=
G G
G G.
7/28/2019 Analytic Geometry Summary Sheet
2/2
Mr. Simonds MTH 253
2 | V e c t o r B a s i c s s t u f f t o b e m e m o r i z e d
The Cross Product
Def 3: If1 2 3, ,u u u u=
G
and1 2 3, ,v v v v=
G
, then
( ) ( ) ( )1 2 3 2 3 3 2 1 3 3 1 1 2 2 11 2 3
i j k
u v u u u u v u v i u v u v j u v u v k
v v v
= = + G G
.
Fact 10: ( )u v v u = G G G G
Fact 11: For two non-zero, non-parallel vectors uG
and vG
, u vG G
is perpendicular to any plane that
is parallel to both uG
and vG
. That is, ( )u v u G G G
and ( )u v v G G G
.
Vector Projections
Def 4 and 5: compuv u
vu
=G
G G
G
Gand ( )proj compu u
v uu uv v
u u u
= = G G
G GG G
G G
G G G
Fact 12: comp 0 proju uv v u> G GG G G
and comp 0 proju uv v u< G GG G G
Additional Fundamental Facts about Vectors, Lines, and Planes
Fact 13: Two non-zero vectors uG
and vG
form a right angle when drawn tail-to-tail if and only if
0u v =G G
. Two such vectors are said to be perpendicular or orthogonal.
Fact 14: Two vectors uG
and vG
are parallel if and only if u k v=G G
for some non-zero scalar k.
Fact 15: A line that is parallel to the vector1 2 3, ,u u u u=
G
and passes through the point
( )0 0 0, ,x y z can be modeled by the vector function ( ) 0 1 0 2 0 3, ,r t x u t y u t z u t = + + +G
and the symmetric equations 0 0 0
1 2 3
x x y y z z
u u u
= = . The vector u
G
is called a
direction vector for the line.
Fact 16: A plane that is perpendicular to the vector 1 2 3, ,u u u u=G
and passes through the point
( )0 0 0, ,x y z can be modeled by the equation ( ) ( ) ( )1 0 2 0 3 0 0u x x u y y u z z + + = .The vector u
G
is called a normal vector for the plane.
Fact 17: The vector1 2 3, ,u u u u=
G
is a normal vector for any plane with an equation of form
1 2 3u x u y u z k + + = .
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