text: # M 408D Lecture 2: HW 02-solutions
  
  60 views13 pages
  Published on 16 Feb 2020
  School
  UT-Austin
  Department
  Mathematics
  Course
  M 408D
  Professor
  oliver (smo998) – HW 02 – rusin – (52850) 1
  This print-out should have 20 questions.
  Multiple-choice questions may continue on
  the next column or page – ﬁnd all choices
  001 10.0 points
  If ais a vector parallel to the xy-plane and
  bis a vector parallel to k, determine ka×bk
  when kak= 3 and kbk= 2.
  1. ka×bk= 0
  2. ka×bk=3
  3. ka×bk= 3
  4. ka×bk= 6 correct
  5. ka×bk=6
  6. ka×bk=32
  7. ka×bk= 32
  Explanation:
  For vectors aand b,
  ka×bk=kakkbksin θ
  when the angle between them is θ, 0 θ < π.
  But θ=π/2 in the case when ais parallel
  to the xy-plane and bis parallel to kbe-
  cause kis then perpendicular to the xy-plane.
  Consequently, for the given vectors,
  ka×bk= 6 .
  keywords: cross product, length, angle,
  002 10.0 points
  A line passes through the point P(2,3,3)
  and is perpendicular to the plane
  x+ 4y+ 3z= 6 .
  At what point Qdoes intersect the xy-
  plane?
  1. Q(3,7,0)
  2. Q(3,0,1)
  3. Q(1,1,0) correct
  4. Q(7,0,3)
  5. Q(1,0,7)
  6. Q(1,1,0)
  Explanation:
  Since the xy-plane is given by z= 0, we
  have to ﬁnd an equation for and then set
  z= 0.
  Now a line passing through a point
  P(a, b, c) and having direction vector vis
  given parametrically by
  r(t) = a+tv,a=ha, b, c i.
  But for , its direction vector will be parallel
  to the normal to the plane
  x+ 4y+ 3z= 6 .
  Thus
  a=h2,3,3i,v=h1,4,3i,
  and so
  r(t) = h2 + t, 3 + 4t, 3 + 3ti.
  In this case, z= 0 when t=1. Conse-
  quently, intersects the xy-plane at
  Q(1,1,0) .
  keywords: line, parametric equations, direc-
  tion vector, point on line, intercept, coordi-
  nate plane
  003 10.0 points
  Which one of the following shaded-regions
  in the plane consists of all points whose polar
  coordinates satisfy the inequalities
  0r < 3,1
  6πθ7
  12π?
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  This preview shows pages 1-3 of the document.
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  oliver (smo998) – HW 02 – rusin – (52850) 2
  1.
  2 4
  2.
  2 4
  3.
  2 4
  4.
  2 4
  correct
  5.
  2 4
  6.
  2 4
  Explanation:
  Using the deﬁnition of polar coordinates
  (r, θ), we see that the region deﬁned by the
  inequalities
  0r < 3,1
  6πθ7
  12π
  is
  2 4
  keywords: polar coordinates, inequalities, po-
  lar graph,
  004 10.0 points
  Find a polar equation for the curve given
  by the Cartesian equation
  4y2=x .
  1. r= 4 csc θtan θ
  2. 4r= sec θtan θ
  3. 4r= sec θcot θ
  4. 4r= csc θcot θcorrect
  5. r= 4 csc θcot θ
  6. r= 4 sec θtan θ
  Explanation:
  We have to substitute for x, y in
  4y2=x
  using the relations
  x=rcos θ , y =rsin θ .
  In this case the Cartesian equation becomes
  4r2sin2θ=rcos θ .
  Unlock document
  
  This preview shows pages 1-3 of the document.
  Unlock all 13 pages and 3 million more documents.
  
  oliver (smo998) – HW 02 – rusin – (52850) 3
  Consequently, the polar form of the equation
  is
  4r= csc θcot θ .
  005 10.0 points
  Use the graph in Cartesian coordinates
  r
  θ
  π2π
  of ras a function of θto determine which
  one of the following is the graph of the corre-
  sponding polar function?
  1.
  2.
  3.
  4.
  5. correct
  6.
  Explanation:
  As the Cartesian graph of rnever crosses
  the θ-axis, there is no value of θin [0,2π]
  at which r= 0; in particular, therefore, the
  polar graph never passes through the origin.
  This eliminates 3 of the graphs, leaving only
  Unlock document
  
  This preview shows pages 1-3 of the document.
  Unlock all 13 pages and 3 million more documents.
  
  75% OFF
  \$9.98/m
  \$39.98/m
  \$39.98
  
  ## or
  
  You will be charged \$119.76 upfront and auto renewed at the end of each cycle. You may cancel anytime under Payment Settings. For more information, see our Terms and Privacy.
explanation: The document is a set of solutions for a mathematics homework assignment (HW 02) from a course at the University of Texas at Austin (UT-Austin). It contains multiple questions and their corresponding answers, each with detailed explanations. The questions cover topics such as vector cross products, parametric equations of lines, polar coordinates, and graphing polar functions.
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  Output rules:
  
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distilabel_metadata.role: system
distilabel_metadata.content: # M 408D Lecture 2: HW 02-solutions
  
  60 views13 pages
  Published on 16 Feb 2020
  School
  UT-Austin
  Department
  Mathematics
  Course
  M 408D
  Professor
  oliver (smo998) – HW 02 – rusin – (52850) 1
  This print-out should have 20 questions.
  Multiple-choice questions may continue on
  the next column or page – ﬁnd all choices
  001 10.0 points
  If ais a vector parallel to the xy-plane and
  bis a vector parallel to k, determine ka×bk
  when kak= 3 and kbk= 2.
  1. ka×bk= 0
  2. ka×bk=3
  3. ka×bk= 3
  4. ka×bk= 6 correct
  5. ka×bk=6
  6. ka×bk=32
  7. ka×bk= 32
  Explanation:
  For vectors aand b,
  ka×bk=kakkbksin θ
  when the angle between them is θ, 0 θ < π.
  But θ=π/2 in the case when ais parallel
  to the xy-plane and bis parallel to kbe-
  cause kis then perpendicular to the xy-plane.
  Consequently, for the given vectors,
  ka×bk= 6 .
  keywords: cross product, length, angle,
  002 10.0 points
  A line passes through the point P(2,3,3)
  and is perpendicular to the plane
  x+ 4y+ 3z= 6 .
  At what point Qdoes intersect the xy-
  plane?
  1. Q(3,7,0)
  2. Q(3,0,1)
  3. Q(1,1,0) correct
  4. Q(7,0,3)
  5. Q(1,0,7)
  6. Q(1,1,0)
  Explanation:
  Since the xy-plane is given by z= 0, we
  have to ﬁnd an equation for and then set
  z= 0.
  Now a line passing through a point
  P(a, b, c) and having direction vector vis
  given parametrically by
  r(t) = a+tv,a=ha, b, c i.
  But for , its direction vector will be parallel
  to the normal to the plane
  x+ 4y+ 3z= 6 .
  Thus
  a=h2,3,3i,v=h1,4,3i,
  and so
  r(t) = h2 + t, 3 + 4t, 3 + 3ti.
  In this case, z= 0 when t=1. Conse-
  quently, intersects the xy-plane at
  Q(1,1,0) .
  keywords: line, parametric equations, direc-
  tion vector, point on line, intercept, coordi-
  nate plane
  003 10.0 points
  Which one of the following shaded-regions
  in the plane consists of all points whose polar
  coordinates satisfy the inequalities
  0r < 3,1
  6πθ7
  12π?
  Unlock document
  
  This preview shows pages 1-3 of the document.
  Unlock all 13 pages and 3 million more documents.
  
  oliver (smo998) – HW 02 – rusin – (52850) 2
  1.
  2 4
  2.
  2 4
  3.
  2 4
  4.
  2 4
  correct
  5.
  2 4
  6.
  2 4
  Explanation:
  Using the deﬁnition of polar coordinates
  (r, θ), we see that the region deﬁned by the
  inequalities
  0r < 3,1
  6πθ7
  12π
  is
  2 4
  keywords: polar coordinates, inequalities, po-
  lar graph,
  004 10.0 points
  Find a polar equation for the curve given
  by the Cartesian equation
  4y2=x .
  1. r= 4 csc θtan θ
  2. 4r= sec θtan θ
  3. 4r= sec θcot θ
  4. 4r= csc θcot θcorrect
  5. r= 4 csc θcot θ
  6. r= 4 sec θtan θ
  Explanation:
  We have to substitute for x, y in
  4y2=x
  using the relations
  x=rcos θ , y =rsin θ .
  In this case the Cartesian equation becomes
  4r2sin2θ=rcos θ .
  Unlock document
  
  This preview shows pages 1-3 of the document.
  Unlock all 13 pages and 3 million more documents.
  
  oliver (smo998) – HW 02 – rusin – (52850) 3
  Consequently, the polar form of the equation
  is
  4r= csc θcot θ .
  005 10.0 points
  Use the graph in Cartesian coordinates
  r
  θ
  π2π
  of ras a function of θto determine which
  one of the following is the graph of the corre-
  sponding polar function?
  1.
  2.
  3.
  4.
  5. correct
  6.
  Explanation:
  As the Cartesian graph of rnever crosses
  the θ-axis, there is no value of θin [0,2π]
  at which r= 0; in particular, therefore, the
  polar graph never passes through the origin.
  This eliminates 3 of the graphs, leaving only
  Unlock document
  
  This preview shows pages 1-3 of the document.
  Unlock all 13 pages and 3 million more documents.
  
  75% OFF
  \$9.98/m
  \$39.98/m
  \$39.98
  
  ## or
  
  You will be charged \$119.76 upfront and auto renewed at the end of each cycle. You may cancel anytime under Payment Settings. For more information, see our Terms and Privacy.
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document is a set of solutions for a mathematics homework assignment (HW 02) from a course at the University of Texas at Austin (UT-Austin). It contains multiple questions and their corresponding answers, each with detailed explanations. The questions cover topics such as vector cross products, parametric equations of lines, polar coordinates, and graphing polar functions.
  
  Label: multiple-question-and-answer