Midterm 1 Solutions
Problem 1
Math 2210

Consider the line L ( t ) = ( 2 + 5t, 2t − 1, 2 ) . Then
L is parallel to the plane 2x − 5y + 18z = 5
L is neither to the plane 4x − 3y − 4z = −29
L is perpendicular to the plane 10x + 4y = -26
You can put in parallel, perpendicular, or neither. Explain how you get the answers.
L ( t ) = ( 2, −1, 2 ) + t ( 5, 2, 0 )
The direction of L is ( 5, 2, 0 )
1. ( 5, 2, 0 ) ⋅ ( 2, −5,18 ) = 0
∴ L is parallel to the plane 2x − 5y + 18z = 5
2. ( 5, 2, 0 ) ⋅ ( 4, −3, −4 ) ≠ 0
( 5, 2, 0 ) × ( 4, −3, −4 ) ≠ 0
∴ L is neither parallel or perpendicular to the plane 4x − 3y − 4z = −29
3. 2 ( 5, 2, 0 ) = (10, 4, 0 )
They are scalar multiples
∴ L is perpendicular to the plane 10x + 4y = -26
Problem 2
Find the arclength of the curve
(
)
r ( t ) = 10t 2 , 2 10t, ln t ,1 ≤ t ≤ 9
(
r ' ( t ) = 20t, 2 10, 1t
)
ds
2
2
= ( 20t ) + 40 + ( 1t )
dt
ds
= 20t + 1t
dt
9
L = ∫ 20t + 1t dt
1
9
10t 2 + ln t 1
Problem 3


Let a= (1, 5, 2 ) and b= ( 7,1, 4 ) be vectors. Find the scalar, vector and orthogonal


projections of b onto a.

Normalize a
 (1, 5, 2 )
a=
30
Scalar Projection
20
  (1, 5, 2 )
a ⋅b =
( 7,1, 4 ) =
30
30
Vector Projection

2 10 4
(1, 5, 2 ) = ⎛⎜⎝ , , ⎞⎟⎠
( a ⋅ b ) a = 20
30
3 3 3
Orthogonal Projection
 
It is definitely not a × b
 
⎛ 2 10 4 ⎞ ⎛ 19 −7 8 ⎞
b − b = ( 7,1, 4 ) − ⎜ , , ⎟ = ⎜ , , ⎟
⎝ 3 3 3⎠ ⎝ 3 3 3⎠
Problem 4
Find an equation of the plane P consisting of all points that are equidistant from
( 3, −2,1) and ( 4, 3, 3) .
Method A
( x − 3)2 + ( y + 2 )2 + ( z − 1)2 = ( x − 4 )2 + ( y − 3)2 + ( z − 3)2
1
1
7 + 5 + 2 = 10
2
2
x + 5y + 2z = 10
x + 5y + 2z =
Method B
( 4, 3, 3) − ( 3, −2,1) = (1, 5, 2 ) = a in problem 3
What is the distance from the origin O= ( 0, 0, 0 ) to the above plane?
Hint: Please refer to problem 3. What is the midpoint of ( 3, −2,1) and
( 4, 3, 3) ?
Method A
Use the formula
1
1 
midpoint= ( 7,1, 4 ) = ( b in problem 3)
2
2
Method B


Nothing but 12 b projects onto a
=
1
2
(scalar projection ) =
10
30
Problem 5
Let P be the plane in space that intersects the x-axis at 10, point A, the
y-axis at 2, point B, and the z-axis at 5, point C.
Find the area of the triangle ABC.
Hint: what is the volume of the tetrahegon OABC, where O= ( 0, 0, 0 ) ?
Method 1
x y z
+ + =1
10 2 5
or
x + 5y + 2z = 10
Exactly the same plane as part one of problem 4.
Volume of tetrahegon OABC
vol= 13 ( area of the base ) ( height )
(
)
⎛ 10 ⎞
vol= 13 5 30 ⎜
⎝ 30 ⎟⎠
Method 2
area AB = ( −10, 2, 0 )
AC = ( −10, 0, 5 )
1
2
AB × AC = (10, 50, 20 ) =
10
30 = 5 30
2