Math 2011 Tutorial 2.pdf

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(1) Equation of Lines in 3D

(a) Vector equations 𝒓 = 𝒓𝟎+ 𝒕𝒗

Where 𝒗 is a vector parallel to the line 𝑳.

(b) Parametric equations

If 𝒗 =< 𝒂, 𝒃, 𝒄 > and 𝒓𝟎=< 𝒙𝟎 , 𝒚𝟎 ,𝒛𝟎 >

𝒙 = 𝒙𝟎+ 𝒂𝒕 , 𝒚 = 𝒚𝟎+ 𝒃𝒕, 𝒛 = 𝒛𝟎+ 𝒄𝒕 .

(c) Symmetric Equations

𝒙−𝒙𝟎

𝒂

=

𝒚−𝒚𝟎

𝒃

=

𝒛−𝒛𝟎

𝒄 .

Example 1 :

Solution :

(2) Equation of a Line Segment

(3) Skew Lines

Example 2 :

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(3) Equation of a plane

(a) Vector Equation of the Plane : 𝒏 ∙ (𝒓 − 𝒓𝟎) = 𝟎 ,

Where 𝒏 is a normal vector to plane and 𝒓 − 𝒓𝟎 is a vector lies on the plane.

(b) Scalar Equation of the Plane thought the point 𝑷𝟎(𝒙𝟎, 𝒚𝟎, 𝒛𝟎) :

𝒂(𝒙 − 𝒙

_𝟎

) + 𝒃(𝒚 − 𝒚

𝟎

) + 𝒄(𝒛 − 𝒛

𝟎

) = 𝟎

Example 2 :

Find an equation of the plane through the point (2, 4, −1) with normal vector 𝑛 =< 2, 3, 4 >. Find the intercepts and sketch the plane. Solution:

Example 3 :

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Example 4 :

Solution:

Example 5 :

Solution:

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Solution:

(5) Distance from a Point to a Line

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(6) Distance between two Lines

Find the distance between the two lines 𝑳𝟏 through point 𝑷𝟏

parallel to vector 𝒗_𝟏 and 𝑳_𝟏 through point 𝑷_𝟐 parallel to vector 𝒗_𝟐.

Solution :

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Example 6 :

Classify the Quadric surface 𝒙𝟐+ 𝟐𝒛𝟐− 𝟔𝒙 − 𝟔𝒚 + 𝟏𝟎 = 𝟎 .

Solution :

(8) The derivative 𝒓′_(𝒕)_{of a vector function}_𝒓(𝒕)_is defined as

(9) Unit Tangent Vector

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Example 7 :

Solution :

(10) Arc-Length

The Arc-length of a space curve has the vector equation

𝒓(𝒕) =< 𝒇(𝒕), 𝒈(𝒕), 𝒉(𝒕) >, 𝒂 ≤ 𝒕 ≤ 𝒃

𝑳 = ∫ √[𝒇𝒃 ′_(𝒕)]𝟐_{+ [𝒈}′_(𝒕)]𝟐_{+ [𝒉}′_(𝒕)]𝟐 𝒂

𝒅𝒕

= ∫ √[𝒅𝒙 𝒅𝒕]

𝟐

+ [𝒅𝒚 𝒅𝒕]

𝟐

+ [𝒅𝒛 𝒅𝒕]

𝟐 𝒃

𝒂 𝒅𝒕

Example 8 :

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(11) Exercises Parallel Lines

1) Is the line through (−𝟒, −𝟔, 𝟏) and (−𝟐, 𝟎, −𝟑)parallel to the line through (𝟏𝟎, 𝟏𝟖, 𝟒) and (𝟓, 𝟑, 𝟏𝟒) ?

Point of Intersection of Lines

2) Determine whether the lines 𝑳𝟏and 𝑳𝟐are parallel, skew, or

intersecting. If they intersect, find the point of intersection

𝑳_𝟏: 𝒙 = 𝟓 − 𝟏𝟐𝒕, 𝒚 = 𝟑 + 𝟗𝒕 , 𝒛 = 𝟏 − 𝟑𝒕

and 𝑳𝟐 : 𝒙 = 𝟑 + 𝟖𝒔, 𝒚 = −𝟔𝒔 , 𝒛 = 𝟕 + 𝟐𝒔 .

3) Determine whether the lines 𝑳𝟏and 𝑳𝟐are parallel, skew, or

intersecting. If they intersect, find the point of intersection.

𝑳𝟏∶ 𝒙−𝟐_𝟏 =𝒚−𝟑_−𝟐 =𝒛−𝟏_−𝟑 and 𝑳𝟐 ∶ 𝒙−𝟑_𝟏 =𝒚+𝟒_𝟑 =𝒛−𝟐_−𝟕 .

Equation of a Plane

4) Find an equation of the plane through the point (𝟓, 𝟑, 𝟓) and with normal vector 𝟐𝒊 + 𝒋 − 𝒌.

5) Find an equation of the plane through the point (𝟐, 𝟒, 𝟔) and parallel to the plane 𝒛 = 𝒙 + 𝒚.

6) Find an equation of the plane through the points (𝟎, 𝟏, 𝟏), (𝟏, 𝟎, 𝟏) and (𝟏, 𝟏, 𝟎) .

7) Find an equation of the plane through the point (𝟏,𝟏_𝟐,𝟏_𝟑) and parallel to the plane 𝒙 + 𝒚 + 𝒛 = 𝟎.

8) Find an equation of the plane that passes through the line of intersection of the planes 𝒙 − 𝒛 = 𝟏 and 𝒚 + 𝟐𝒛 = 𝟑 and is perpendicular to the plane 𝒙 + 𝒚 − 𝟐𝒛 = 𝟏 .

Intersection Point of a Line and a Plane

9) Find the point at which the line intersects the given plane

𝒙 = 𝟏 + 𝟐𝒕, 𝒚 = 𝟒𝒕, 𝒛 = 𝟐 − 𝟑𝒕 ; 𝒙 + 𝟐𝒚 − 𝒛 + 𝟏 = 𝟎 .

10) Where does the line through (𝟏, 𝟎, 𝟏) and(𝟒, −𝟐, 𝟐) intersect the plane 𝒙 + 𝒚 + 𝒛 = 𝟔 ?

11) Find the distance from the point (𝟏, −𝟐, 𝟒) to the given plane

𝟑𝒙 + 𝟐𝒚 + 𝟔𝒛 = 𝟓 .

12) Find the distance from the point (−𝟔, 𝟑, 𝟓) to the plane

𝒙 − 𝟐𝒚 − 𝟒𝒛 = 𝟖 .

13) Find the distance between the parallel planes

𝟐𝒙 − 𝟑𝒚 + 𝒛 = 𝟒 𝐚𝐧𝐝 𝟒𝒙 − 𝟔𝒚 + 𝟐𝒛 = 𝟑 .

14) Find the distance between the parallel planes

𝟔𝒛 = 𝟒𝒚 − 𝟐𝒙 𝒂𝒏𝒅 𝟗𝒛 = 𝟏 − 𝟑𝒙 + 𝟔𝒚 .

15) Show that the distance between the parallel planes

𝒂𝒙 + 𝒃𝒚 + 𝒄𝒛 + 𝒅𝟏= 𝟎 and 𝒂𝒙 + 𝒃𝒚 + 𝒄𝒛 + 𝒅𝟐= 𝟎 is

𝑫 = |𝒅𝟏− 𝒅𝟐| √𝒂𝟐_{+ 𝒃}𝟐_{+ 𝒄}𝟐

Distance Between Two Skew Lines

16) Find the distance between the skew lines with parametric Equations

𝒙 = 𝟏 + 𝒕 , 𝒚 = 𝟏 + 𝟔𝒕 , 𝒛 = 𝟐𝒕 , and 𝒙 = 𝟏 + 𝟐𝒔 , 𝒚 = 𝟓 + 𝟏𝟓𝒔 ,

𝒛 = −𝟐 + 𝟔𝒔 .

17) Let 𝑳𝟏 be the line through the points (𝟏, 𝟐, 𝟔) and (𝟐, 𝟒, 𝟖). Let 𝑳𝟐 be

the line of intersection of the planes 𝝅𝟏 and 𝝅𝟐 , where 𝝅𝟏 is the

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(𝟑, 𝟐, −𝟏) , (𝟎, 𝟎, 𝟏) and (𝟏, 𝟐, 𝟏). Calculate the distance between 𝑳𝟏

and 𝑳𝟐 .

Identify Quadric Surfaces

18) Sketch and identify the surface 𝒙𝟐 = 𝒚𝟐+ 𝟒𝒛𝟐 .

19) Sketch and identify the surface 𝟐𝟓𝒙𝟐+ 𝟒𝒚𝟐+ 𝒛𝟐 = 𝟏𝟎𝟎 .

20) Sketch and identify the surface 𝟒𝒙𝟐+ 𝟒𝟗𝒚𝟐+ 𝒛𝟐= 𝟎 .

(For Q21-23)Reduce the equation to one of the standard forms, classify the surface, and sketch it.

21) 𝟒𝒙𝟐− 𝒚 + 𝟐𝒛𝟐 = 𝟎 .

22) 𝟒𝒙𝟐+ 𝒚𝟐+ 𝟒𝒛𝟐− 𝟒𝒚 − 𝟐𝟒𝒛 + 𝟑𝟔 = 𝟎 .

23) 𝒙𝟐− 𝒚𝟐+ 𝒛𝟐− 𝟐𝒙 + 𝟐𝒚 + 𝟒𝒛 + 𝟐 = 𝟎 .

24) Sketch the region bounded by the paraboloids 𝒛 = 𝒙𝟐+ 𝒚𝟐 and

𝒛 = 𝟐 − 𝒙𝟐_{− 𝒚}𝟐_.

25) Show that the curve of intersection of the surfaces

𝒙𝟐+ 𝟐𝒚𝟐− 𝒛𝟐+ 𝟑𝒙 = 𝟏 and 𝟐𝒙𝟐+ 𝟒𝒚𝟐− 𝟐𝒛𝟐− 𝟓𝒚 = 𝟎 lies in a plane.

26) Sketch the curve with the vector equation 𝒓(𝒕) =< 𝒔𝒊𝒏𝒕, 𝒕 >. Indicate with an arrow the direction in which increases.

27) Sketch the curve with the vector equation

𝒓(𝒕) =< 𝒔𝒊𝒏𝝅𝒕, 𝒕, 𝒄𝒐𝒔𝝅𝒕 >.

Indicate with an arrow the direction in which increases.

𝒓(𝒕) =< 𝟏, 𝒄𝒐𝒔𝒕, 𝟐𝒔𝒊𝒏𝒕 >.

29) Sketch the curve with the vector equation 𝒓(𝒕) = 𝒕𝟐𝒊 + 𝒕𝒋 + 𝟐𝒌. Indicate with an arrow the direction in which increases.

𝒓(𝒕) = 𝒄𝒐𝒔𝒕 𝒊 − 𝒄𝒐𝒔𝒕 𝒋 + 𝒔𝒊𝒏𝒕 𝒌.

31) Show that the curve with parametric equations 𝒙 = 𝒕𝒄𝒐𝒔𝒕 ,

𝒚 = 𝒕 𝒔𝒊𝒏𝒕 , 𝒛 = 𝒕 lies on the cone 𝒛𝟐= 𝒙𝟐+ 𝒚𝟐 , and use this fact to help sketch the curve.

32) At what points does the helix 𝒓(𝒕) =< 𝒔𝒊𝒏𝒕, 𝒄𝒐𝒔𝒕, 𝒕 > intersect the sphere 𝒙𝟐+ 𝒚𝟐+ 𝒛𝟐= 𝟓 ?

33) Show that the curve with parametric equations

𝒙 = 𝒕𝟐_,_{𝒚 = 𝟏 − 𝟑𝒕 , 𝒛 = 𝟏 + 𝒕}𝟑

passes through the points (𝟏, 𝟒, 𝟎) and (𝟗, −𝟖, 𝟐𝟖) but not through the point (𝟒, 𝟕, −𝟔).

34) Find a vector function that represents the curve of intersection of the surfaces :

the cylinder 𝒙𝟐+ 𝒚𝟐 = 𝟒 and the surface 𝒛 = 𝒙𝒚.

35) Find a vector function that represents the curve of intersection of the cone

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36) Find a vector function that represents the curve of intersection of the paraboloid 𝒛 = 𝟒𝒙𝟐+ 𝒚𝟐 and the parabolic cylinder 𝒚 = 𝒙𝟐.

37) Find a vector function that represents the curve of intersection of the hyperboloid 𝒛 = 𝒙𝟐− 𝒚𝟐 and the cylinder 𝒙𝟐+ 𝒚𝟐 = 𝟏.

38) Try to sketch by hand the curve of intersection of the circular cylinder 𝒙𝟐+ 𝒚𝟐 = 𝟒 and the parabolic cylinder 𝒛 = 𝒙𝟐. Then find parametric equations for this curve and use these equations and a computer to graph the curve.

Applications

39) If two objects travel through space along two different curves, it’s often important to know whether they will collide. (Will a missile hit its moving target? Will two aircraft collide?) The curves might intersect, but we need to know whether the objects are in the same position at the same time. Suppose the trajectories of two particles are given by the vector functions

𝒓𝟏(𝒕) =< 𝒕𝟐, 𝟕𝒕 − 𝟏𝟐, 𝒕𝟐>

and 𝒓𝟐(𝒕) =< 𝟒𝒕 − 𝟑, 𝒕𝟐, 𝟓𝒕 − 𝟔 > for 𝒕 ≥ 𝟎.

Do the particles collide?

40) Two particles travel along the space curves 𝒓𝟏(𝒕) =< 𝒕, 𝒕𝟐, 𝒕𝟑 >

and 𝒓𝟐(𝒕) =< 𝟏 + 𝟐𝒕, 𝟏 + 𝟔𝒕, 𝟏 + 𝟏𝟒𝒕 > .

Do the particles collide? Do their paths intersect?

Tangent Vectors and Tangent Lines

41) (a) Make a large sketch of the curve described by the vector function 𝒓(𝒕) =< 𝒕𝟐, 𝒕 >, 𝟎 ≤ 𝒕 ≤ 𝟐, and draw the vectors

𝒓(𝟏), 𝒓(𝟏. 𝟏), and 𝒓(𝟏. 𝟏) − 𝒓(𝟏).

(b) Draw the vector 𝒓′(𝒕) starting at (𝟏, 𝟏), and compare it with the vector 𝒓(𝟏.𝟏)−𝒓(𝟏)_𝟎.𝟏 . Explain why these vectors are so close to each other in length and direction.

42) (a) Sketch the plane curve with the vector equation

𝒓(𝒕) =< 𝒕 − 𝟐, 𝒕𝟐_{+ 𝟏 >}_,_{𝒕 = −𝟏}_.

(b) Find 𝒓′(𝒕) .

(c) Sketch the position vector 𝒓(𝒕) and the tangent vector

𝒓′_(𝒕)_for_{𝒕 = −𝟏}_.

43) Find the unit tangent vector at the point with the given value of the parameter .

(a) 𝒓(𝒕) =< 𝒕𝟑+ 𝟑𝒕, 𝒕𝟐+ 𝟏, 𝟑𝒕 + 𝟒 >, 𝒕 = 𝟏 .

(b) 𝒓(𝒕) = 𝒔𝒊𝒏𝟐𝒕 𝒊 + 𝒄𝒐𝒔𝟐𝒕 𝒋 + 𝒕𝒂𝒏𝟐𝒕 𝒌, 𝒕 =𝝅_𝟒 .

44) Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point.

(a) 𝒙 = 𝟏 + 𝟐√𝒕 , 𝒚 = 𝒕𝟑− 𝒕 , 𝒛 = 𝒕𝟑+ 𝒕 ; (𝟑, 𝟎, 𝟐) ,

(b) 𝒙 = √𝒕𝟐+ 𝟑 , 𝒚 = 𝒍𝒏(𝒕𝟐+ 𝟑) , 𝒛 = 𝒕 ; (𝟐, 𝒍𝒏𝟒, 𝟏) .

45) Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. Illustrate by graphing both the curve and the tangent line on a common screen.

(a) 𝒙 = 𝒕 , 𝒚 = 𝒆−𝒕 , 𝒛 = 𝟐𝒕 − 𝒕𝟐 ; (𝟎, 𝟏, 𝟎) ,

(b) 𝒙 = 𝟐𝒄𝒐𝒔𝒕 , 𝒚 = 𝟐𝒔𝒊𝒏𝒕 , 𝒛 = 𝟒𝒄𝒐𝒔𝟐𝒕 ; (√𝟑, 𝟏, 𝟐) ,