Review for Chapter 14
1. Operation of vecters.
The addition and the multiplication of vectors behave pretty much the same way as the
operations of numbers with some exceptions (see (7) of page 633, (9) of page 634 and Theorem
3 of page 643). The following are some reminders:
(i) The dot product of two vectors outputs a number.
(ii) The scalar product and the vector product output vectors.
(iii) The vector product of two vectors is not commutative.
2. The distance formula and its relation to dot products.
If ~r has its head in (x1, y1, z1) and tail at (x2, y2, z2), then ~r =< x1−x2, y1−y2, z1−z2>,
and
|~r|2=~r •~r = (x1−x2)2+ (y1−y2)2+ (z1−z2)2.
3. Important facts: (the angle θbelow is the angle between the two vectors ~a and ~
b)
(i) ~a •~
b=|~a||~
b|cos θ, and so ~a and ~
bare perpendicular if and only if ~a •~
b= 0.
(ii) |~a ×~
b|2=|~a|2|~
b|2sin2θ, and so ~a and ~
bare parallel to each other if and only if ~a ×~
b=~
0,
the zero vector.
(iii) ~a ×~
bis perpendicular to both ~a and ~
b.
4. (Sction 14.1) Direction angles and numbers.
Let ~r =< x, y, z >. Let the angles between ~r and the x,yand zaxes be α,βand γ,
respectively. Then these angles are the direction angles of ~r and the direction numbers are
cos α=~r •~
i
|~r|,cos β=~r •~
j
|~r|,cos γ=~r •~
k
|~r|.
5. (Vector components, Section 14.1) The component of ~a along ~
bis
Comp~
b~a =~a •~
b
|~
b|.
6. (Equations of a line, Section 14.3) Let Lbe a line in space that is parallel to ~n =< a, b,c >
and passes through (x0, y0, z0). Then the parametric equation of Lis
~
r(t) =< at +x0, bt +y0, ct +z0>,
and the symmetric equations of Lis
x−x0
a=y−y0
b=z−z0
c.
7. (Facts about planes, Section 14.3) The plane passes (x0, y0, z0) with normal vector ~n =<
a, b, c > has equation
< a, b, c > •< x −x0, y −y0, z −z0>= 0.
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