Math 54 Exam 3 Review

8/20/2019 Math 54 Exam 3 Review

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Schedule Coverage Sample Questions

Reminder

THIRD LONG EXAMINATION

21 October 2014, Tuesday

Discussion Class Room and Time

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Coverage

3; three dimensional coordinate system

Distance between two points, midpoint of a line segment

Spheres

Vectors in 3

Properties

(e.g. magnitude, direction angles, parallel and perpendicular vectors)

Vector operations

(addition/subtraction, multiplication by a scalar, dot product, crossproduct)

Applications (e.g. area, volume)

Lines and planes in three-dimensional space

Representations (equation of a plane, vector/symmetric/parametric

equations of lines)

Intersections

Distances

More surfaces

Cylinders

Surfaces of revolution

Quadric surfaces2/19

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Sample Questions

Write TRUE if the statement is always true, and FALSE if otherwise.

1. A vector of length 3 that is parallel toa is 3

a .

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a . FALSE

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a . FALSE

2. The direction angles of −6

3 ˆ +6k̂ are α= π2

, β= 5π6

, and γ= π3

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S h d l C S l Q ti

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Sample Questions



a . FALSE


3 ˆ +6k̂ are α= π2

, β= 5π6

, and γ= π3

TRUE

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Sample Questions



a . FALSE


3 ˆ +6k̂ are α= π2

, β= 5π6

, and γ= π3

TRUE

3. The vector equationr = 〈9,− 32 ,−6〉+〈−1,4,6〉t and the symmetric

equations ℓ2 : 8−x = 2 y

−5

8 = z

6 represent the same line.

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Sample Questions



a . FALSE


3 ˆ +6k̂ are α= π2

, β= 5π6

, and γ= π3

TRUE



−5

8 = z

6 represent the same line. TRUE

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Sample Questions



a . FALSE


3 ˆ +6k̂ are α= π2

, β= 5π6

, and γ= π3

TRUE



−5

8 = z

6 represent the same line. TRUE4. The graph of the equation 4x 2 − y 2−9z 2+1= 0 is a hyperboloid of one

sheet.

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Sample Questions



a . FALSE


3 ˆ +6k̂ are α= π2

, β= 5π6

, and γ= π3

TRUE



−5

8 = z


sheet. TRUE

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Sample Questions



a . FALSE


3 ˆ +6k̂ are α= π2

, β= 5π6

, and γ= π3

TRUE



−5

8 = z


sheet. TRUE

5. If a ,b , andc are non-zero vectors, thena · (b ×c )= (c ×b ) ·a .

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Sample Questions



a . FALSE


3 ˆ +6k̂ are α= π2

, β= 5π6

, and γ= π3

TRUE


equations ℓ2 : 8−x = 2 y −5

8 = z


sheet. TRUE

5. If a ,b , andc are non-zero vectors, thena · (b ×c )= (c ×b ) ·a . FALSE

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Sample Questions



a . FALSE


3 ˆ +6k̂ are α= π2

, β= 5π6

, and γ= π3

TRUE


equations ℓ2 : 8−x = 2 y −5

8 = z


sheet. TRUE

5. If a ,b , andc are non-zero vectors, thena · (b ×c )= (c ×b ) ·a . FALSE6. If u

×v

=0 then eitheru

=0 orv

=0.

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Sample Questions



a . FALSE


3 ˆ +6k̂ are α= π2

, β= 5π6

, and γ= π3

TRUE

3. The vector equationr = 〈9,− 32 ,−6〉+〈−1,4,6〉t and the symmetricequations ℓ

2 : 8

−x =

2 y −58 =

z

6 represent the same line. TRUE

4. The graph of the equation 4x 2 − y 2−9z 2+1= 0 is a hyperboloid of onesheet. TRUE

5. If a ,b , andc are non-zero vectors, thena · (b ×c )= (c ×b ) ·a . FALSE6. If u

×v

=0 then eitheru

=0 orv

=0. FALSE

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Sample Questions

Given:a = 〈−3,4,

15〉 andb =−4ı̂ +2 ˆ

1. Find m so that 〈m ,−7,0〉 is parallel to

b .

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Sample Questions

Given:a = 〈−3,4,

15〉 andb =−4ı̂ +2 ˆ

2. Find the angle betweena and

b .

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Sample Questions

Given: A (−1,2,−5), B (3,10,−1), C (1,8,−7), and D (−2,5,0)

1. Find the equation of the sphere having AB as a diameter.

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Sample Questions

Given: A (−1,2,−5), B (3,10,−1), C (1,8,−7), and D (−2,5,0)

2. Find the symmetric equations of the line containing C and D .

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Sample Questions

Given: A (−1,2,−5), B (3,10,−1), C (1,8,−7), and D (−2,5,0)

3. Find the area of △ ABC .

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Sample Questions

Given: A (−1,2,−5), B (3,10,−1), C (1,8,−7), and D (−2,5,0)

4. Find the volume of the parallelepiped having the edges AB , AC , and AD .

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Sample Questions

Express the vectorv = 〈−2,4,5〉 as a sum of two vectorsu andw such thatu is parallel to the vector 6 ı̂

−2 ˆ andw

⊥u .

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S l Q i

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Sample Questions

Given: ℓ1 :

x = 1 y =

1

2 +t

z =−3t , ℓ2 :

x = 2s y =−

1+

2s

z = 6−9s

1. Find the distance of the point−1, 3

2, 0

from ℓ1.

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S l Q i

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Sample Questions

Given: ℓ1 :

x = 1 y =

1

2 +t

z =−3t , ℓ2 :

x = 2s y

=−1

+2s

z = 6−9s 2. Find the intersection of ℓ1 and ℓ2.

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Sample Questions

Given: ℓ1 :

x = 1 y =

1

2 +t

z =−3t , ℓ2 :

x = 2s y

=−1

+2s

z = 6−9s 3. Find the equation of the plane π containing both lines.

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S l Q ti



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Sample Questions

Given: ℓ1 :

x = 1 y

= 1

2 +t

z =−3t , ℓ2 :

x = 2s y

=−1

+2s

z = 6−9s 4. Sketch the plane π in the first octant.

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Sample Questions

Given: ℓ1 :

x = 1 y

= 1

2 +t

z =−3t , ℓ2 :

x = 2s y

=−1

+2s


π : 3x

+6 y

+2z

=6

x -intercept:

y -intercept:

z -intercept:

x y

z

1

2

3

1

2

1

2

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Sample Questions

Given: ℓ1 :

x = 1 y

= 1

2 +t

z =−3t , ℓ2 :

x = 2s y

=−1

+2s


π : 3x

+6 y

+2z

=6

x -intercept: 2

y -intercept:

z -intercept:

x y

z

1

2

3

1

2

1

2

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Sample Questions

Given: ℓ1 :

x = 1 y

= 1

2 +t

z =−3t , ℓ2 :

x = 2s y

=−1

+2s


π : 3x

+6 y

+2z

=6

x -intercept: 2

y -intercept: 1

z -intercept:

x y

z

1

2

3

1

2

1

2

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Sample Questions

Given: ℓ1 :

x = 1 y

= 1

2 +t

z =−3t , ℓ2 :

x = 2s y

=−1

+2s


π : 3x

+6 y

+2z

=6

x -intercept: 2

y -intercept: 1

z -intercept: 3

x y

z

1

2

3

1

2

1

2

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Sample Questions

Given: ℓ1 :

x = 1 y

= 1

2 +t

z =−3t , ℓ2 :

x = 2s y

=−1

+2s


π : 3x

+6 y

+2z

=6

x -intercept: 2

y -intercept: 1

z -intercept: 3

x y

z

1

2

3

1

2

1

2

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Sample Questions

Given: π1 : 2x −2 y +z = 3, π2 : 7−4x +4 y −2z = 01. Show that the two planes are parallel.

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Sample Questions

Given: π1 : 2x −2 y +z = 3, π2 : 7−4x +4 y −2z = 02. Find the distance between the two given planes

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Sample Questions

Sketch the graph of the equation x 2+4z 2 −2x −8z +1 = 0

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The graph of the equation is a

cylinder

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p Q



cylinder generated by the ellipse

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p Q




(x −1)2

4+ (z −1)2 = 1

on the xz -plane.

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p Q




(x −1)2

4+ (z −1)2 = 1

on the xz -plane.

x y

z

1

2

3

1

2

3

1

23

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p Q




(x −1)2

4+ (z −1)2 = 1

on the xz -plane.

x y

z

1

2

3

1

2

3

1

23

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(x −1)2

4+ (z −1)2 = 1

on the xz -plane.

x y

z

1

2

3

1

2

3

1

23

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(x −1)2

4+ (z −1)2 = 1

on the xz -plane.

x y

z

1

2

3

1

2

3

1

23

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Given: z = 3x 2+3 y 21. Identify its traces on the coordinate planes and sketch its graph.

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The graph of the equation is an

elliptic paraboloid, a circular

paraboloid in particular.

xy -trace:

xz -trace:

yz -trace:

x y

z

1

2

3

1

2

1

2

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xy -trace: (0,0,0) (point)

xz -trace:

yz -trace:

x y

z

1

2

3

1

2

1

2

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xz -trace:

yz -trace:

x y

z

1

2

3

1

2

1

2

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xz -trace: z = 3x 2 (parabola) yz -trace:

x y

z

1

2

3

1

2

1

2

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xz -trace: z = 3x 2 (parabola) yz -trace:

x y

z

1

2

3

1

2

1

2

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xz -trace: z = 3x 2 (parabola) yz -trace: z = 3 y 2 (parabola)

x y

z

1

2

3

1

2

1

2

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xz -trace: z = 3x 2 (parabola) yz -trace: z = 3 y 2 (parabola)

x y

z

1

2

3

1

2

1

2

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xz -trace: z = 3x 2 (parabola) yz -trace: z = 3 y 2 (parabola)trace on z = 3:

x y

z

1

2

3

1

2

1

2

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xz -trace: z = 3x 2 (parabola) yz -trace: z = 3 y 2 (parabola)trace on z = 3: x 2+ y 2 = 1

(circle)

x y

z

1

2

3

1

2

1

2

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(circle)

x y

z

1

2

3

1

2

1

2

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(circle)

x y

z

1

2

3

1

2

1

2

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Given: z = 3x 2+3 y 22. Find the equation of a curve in the yz -plane such that if the curve is

revolved about the z -axis, the result will be given surface.

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Documents

Math 54 Exam 3 Review