241C -Study Guide for Test #1
Test Date: Sept. 26, 2005
Do any four problems. You may do one, two , or three problems for extra credit (Score/2)
Problem #1 Given vectors A , B and C:
(a)-(i)-[5 pts.] Find scalars a and b such that C = aA + bB.
(a)-(ii)-[5 pts.] Draw arrows representing 2A, A + B and A - C.
(a)-(iii)-[5 pts.] Draw an arrow representing A + B + C.
(b)-[5 pts.] Calculate the cosine of the angle between PQ and PR if
P =(x1, y1, z1), Q = (x2, y2, z2), R = (x3, y3, z3).
(c)-[5 pts.] Show that two given vectors, A and B, are perpendicular.
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Problem #2 Given A and B:
(a)-(i)-[6 pts.] Find Pr ojBA .
(a)-(ii)-[3 pts.] Find A - Pr ojBA .
(a)-(ii)-[3 pts.] Show that Pr ojBA and A - Pr ojBA are perpendicular.
(a)-(iii)-[3 pts.] Express A in terms of Pr ojBA and A - Pr ojBA .
(b)-[10 pts.] Given a vector A and mutually perpendicular unit vectors
u 1 , u2 and u 3 find scalars x, y, and z such that A = x u 1 + y u2 + z u 3 .
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Problem #3
(a)-[8 pts.] Find the area of the parallelogram determined by given vectors vectors A and B.
(b)-[9 pts.] Find a vector perpendicular to the plane determined by three given points.
(c)-[8 pts.] Find the volume of the parallelepiped determined by three given vectors.
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Problem #4
(a)-[13 pts.] Find the vector and parametric equations of the line determined by two given points.
(b)-[12 pts.] Find the distance of a given point S from the line determined in part (a) (Memorize
and know how to apply the formula given in prob. #39 on page 821).
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Problem #5
(a)-[13 pts.] Find the equation of the plane determined by three given points.
(b)-[12 pts.] Find the distance of a given point S from a given plane ) (Memorize
and know how to apply formula 9 on page 828. Also memorize and
and know how to apply the formula given in prob. #40 on page 828. ).
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Problem #6
(a)-(i)-[6 pts.] & (a)-(ii)-[7 pts.]
Sketch the graph of Ax2 + By2 + C z2 = D where A, B and C are given constants.
Hint: To assist you in answering (a)-(i) and (a)-(ii) you should do the following:
1. Find vertical and horizontal cross sections of the given surface, in particular cross
sections by the coordinate planes.
2. Determine any symmetry of the surface with respect to the origin, to any of the
coordinate axes, or to any of the coordinate planes.
3. Determine if the surface has one sheet or two.
4. Find the intercepts of the surface with the coordinate axes.
(b)-(i) [8 pts.] & (b)-(ii)-[4 pts.]
Describe a given surface of the form z = ax2+ by2 + c. Is it a surface of revolution?
Is it shaped like a saddle?
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Problem #7
(a)-[6 pts.] Express the surface represented by an equation of the form z =f(x, y) in cylindrical coordinates,
and sketch it. Briefly describe the surface.
2
(b)-[3 pts.] Express the surface represented by an equation of the form (x  x1)2  (y  y1)2  z  z1  r 2
in spherical coordinates. Briefly describe the surface.
(c)-(i) [8 pts.] & (c)-(ii) [8 pts.]
Express the surface represented by the equation r  1 / (p  qcos  )
in rectangular coordinates. Here p and q are given constants. Bring the
2
2
equation into the form ax  by  cx  dy  e  0 and briefly describe the surface.
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C.O. Bloom