MTH 253 Test 3 Material – Vectors and Lines and Planes, oh my
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The following facts and formulae are things you need to memorize.
Basic Vector Language and Vector Arithmetic
Fact 1: The three standard unit vectors are ˆ1, 0, 0i=, ˆ0,1,0j= and ˆ0,0,1k=. The
vector 123
,,uuuu=
G can also be written as 12 3
ˆ
ˆˆ
uuiujuk=+ +
G
. Consequently, the
components of the vector u
G
; 1
u, 2
u, and 3
u; are called, respectively, the ˆ
i−component,
the ˆ
j−component, and the ˆ
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 both
could
each be drawn in
their entirety along the same line. That is, two non-zero are 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
could
be 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
could
be 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 123
,,uuuu=
G is 222
123
uuuu=++
G
. In 2-dimensions and
3-dimenstions this quantity is commonly referred to as the length of the vector. In
applied problems this quantity is commonly referred to as the magnitude of the vector.
Fact 7: A vector, v
G, is called a unit vector if and only if 1v
=
G
. Unit vectors are generally
annotated thusly: ˆ
v.
Fact 8: For a non-zero vector v
G, the vector ˆv
uv
=
G
G
is the unit vector that points in the same
direction as v
G. The process of dividing a vector by its length is called normalizing the
vector.
