Worksheet 2
Part A
1. Draw a picture as follows: Draw the origin O and another point P . Draw a line that passes
through the point P . Draw a vector v that starts at P and is parallel to the line. Draw
another point on the line and label it Q = (x, y, z).
Suppose for problems 2–4 that P = (2, 3, 4) and v = h2, 3, 6i.
2. What are the components of the vector from P to Q?
3. Recall that two vectors are parallel if one is a scalar multiple of the other. Use this idea to
!
write an equation that represents the statement, “P Q is parallel to v.”
4. Solve your equation for hx, y, zi. You have just found an equation for the line through P
parallel to v.
5. Repeat steps 2–4 for P = (a, b, c) and v = hv1 , v2 , v3 i.
Part B
6. Draw a picture as follows: Draw a plane containing a point P . Draw a vector n that starts
at P and is perpendicular to the plane. Draw any other point on the plane and label it
Q = (x, y, z).
Suppose for problems 7–8 that P = (2, 3, 4) and n = h1, 5, 4i.
7. Recall that two vectors are perpendicular if their dot product is zero. Use this idea to write
!
an equation that represents the statement, “P Q is perpendicular to n.”
8. Simplify your equation so that the variables appear on one side and the other side is constant.
You have just found an equation for the plane through P perpendicular to n.
9. Repeat steps 7–8 for P = (a, b, c) and n = hn1 , n2 , n3 i.
1
Part A
>
3,6
I
1)
0
↳
Z
^
g
g-
PIT
if
••8
@(
9%3147
'
✗
•
if
>
^
"
-1
2)
@
=P
=
( x -2
,
y -3
,
-47
-2
Q=Cxiy,z)
be
&
P=(2
3) FÉ
parallel
42T
:
{
,
-3T
,
2t=x -2
-
3t=y -3
6t=Z -4
to
let > =
{
✗
( x -2
F
,y
1-+0
-3
,
2- =
*=
6++4
-
parametric equation
of
the
lire
.
3,
treat
such
4)
tF=PÑ
.
,
-2-4 >
=2t -12
y= -3-1+3
,
{
⇐
1-
=
×÷→ssmTautÉ°
Y -3
3
t=z÷
"
Part B
nñ
:
6)
>•Q
p•
>
PQ
is
pot
•
PÑ
n→
=
=
in
perpendicular
to
Ñ
=D
Cx -2
,
y -3
51,5 , -47
,
2--47