Solutions - SLC Home Page

Math 105
Semester Review - Solutions
b) Find the angle between
AB and AC .
ABiAC
−4
cosθ
= =
θ ≈ 111
AB AC 5 25
c) Find the volume of the tetrahedron ABCD.
1 0 2
3 0 −4 3
1 1 1 1
8
V =
6
ABi ( AC× AD)
=
6
− 4 3 0 =
6
1 − 0+ 2 =
6
6+
10 =
1 2 −3 1 3
−3 1 2
d) Find the equation of the line (in parametric form) passing through D and parallel to
AB .
⎧x
= − 1+
t
⎪
u = AB=
(1, 0, 2)
l: ⎨ y = 0
⎪ ⎩z
= 5 + 2t
e) Find the equation of the plane (in general form) parallel to AB and AC , and passing
through D.
i j k
n = AB× AC = 1 0 2 = −6, −8,3
−4 3 0
( ) ( ) ( )
−6x− 8x+ 3z
=−6 −1 − 8 0 + 3 5 = 21
π :6x+ 8y− 3z
=− 21
(
f) Find the equation of the plane perpendicular to AC and passing through D.
n = AC = −4,3,0
− 4x+ 3y =−4 − 1 + 3 0 + 0 5 = 4
( )
)
( ) ( ) ( )
π :4x− 3y
=− 4
x −1
2y
+ 1
19. Consider the plane π : 2x
+ y − 5z
+ 1 = 0 and the line L : = = 3 − z
3 4
a) Find the equation of the line (in symmetric form) perpendicular to π and passing
through P(1,1,-3).
x− 1 z+
3
u = (2,1, −5)
= y − 1 =
2 − 5
b) Find the equation of the line (in parametric form) parallel to L and passing through
P(1,1,-3).
⎧x
= 1+
3t
⎪
u = (3, 2, −1)
⎨y
= 1 + 2t
⎪
⎩z
= − 3 − t
Winter 2006 Martin Huard 14