3. Interpret the graph of x = 1 in the contexts of (a) a number line (b) 2-space (c) 3-space

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1 MA2: Prepared by Dr. Archara Pacheenburawana Exercise Chapter 3 Exercise 3.. A cube of side 4 has its geometric center at the origin and its faces parallel to the coordinate planes. Sketch the cube and give the coordinates of the corners. 2. Suppose that a box has its faces parallel to the coordinate planes and the points (4,2, 2) and ( 6,,) are endpoints of a diagonal. Sketch the box and give the coordinates of the remaining six corners. 3. Interpret the graph of x = in the contexts of (a) a number line (b) 2-space (c) 3-space 4. Findthecenterandradiusofthespherethathas(, 2,4)and(3,4, 2)asendpoints of a diameter. 5. Show that (4,5,2), (,7,3), and (2,4,5) are vertices of an equilateral triangle. 6. (a) Show that (2,,6), (4,7,9), and (8,5, 6) are the vertices of a right triangle. (b) Which vertex is at the 90 angle? (c) Find the area of the triangle. 7. Find equations of two spheres that are centered at the origin and are tangent to the sphere of radius centered at (3, 2,4). 8 3 Describe the surface whose equation is given. 8. x 2 +y 2 +z 2 +0x+4y +2z 9 = 0 9. x 2 +y 2 +z 2 y = x 2 +2y 2 +2z 2 2x 3y +5z 2 = 0. x 2 +y 2 +z 2 +2x 2y +2z +3 = 0 2. x 2 +y 2 +z 2 3x+4y 8z +25 = 0 3. x 2 +y 2 +z 2 2x 6y 8z + = 0 4. In each part, sketch the portion of the surface that lies in the first octant. (a) y = x (b) y = z (c) x = z 5. In each part, sketch the graph of the equation in 3-space. (a) x = (b) y = (c) z = 6. In each part, sketch the graph of the equation in 3-space. (a) x 2 +y 2 = 25 (b) y 2 +z 2 = 25 (c) x 2 +z 2 = 25

2 MA2: Prepared by Dr. Archara Pacheenburawana 2 7. In each part, sketch the graph of the equation in 3-space. (a) x = y 2 (b) z = x 2 (c) y = z Sketch the surface in 3-space. 8. y = sinx 9. y = e x 20. z = y 2 2. z = cosx 22. 2x+z = x+3y = x 2 +9z 2 = z = 3 x 26. y 2 4z 2 = yz = 28. If a bug walks on the sphere x 2 +y 2 +z 2 +2x 2y 4z 3 = 0 how close and how far can it get from the origin? 29. Describe the set of all points in 3-space whose coordinates satisfy the inequality x 2 +y 2 +z 2 2x+8z Describe the set of all points in 3-space whose coordinates satisfy the inequality y 2 +z 2 +6y 4z > The distance between a point P(x,y,z) and the point A(, 2,0) is twice the distance between P and the point B(0,,). Show that the set of all such points is a sphere, and find the center and radius of the sphere. Exercise Sketch the vectors with their initial points at the origin.. (a) 2,5 (b) 5, 4 (c) 2,0 (d) 5i+3j (e) 3i 2j (f) 6j 2. (a) 3,7 (b) 6, 2 (c) 0, 8 (d) 4i+2j (e) 2i j (f) 4i 3. (a), 2,2 (b) 2,2, (c) i+2j+3k (d) 2i+3j k 4. (a),3,2 (b) 3,4,2 (c) 2j k (d) i j+2k

3 MA2: Prepared by Dr. Archara Pacheenburawana Find the components of the vector P P (a) P (3,5), P 2 (2,8) (b) P (7, 2), P 2 (0,0) (c) P (5, 2,), P 2 (2,4,2) 6. (a) P ( 6, 2), P 2 ( 4, ) (b) P (0,0,0), P 2 (,6,) (c) P (4,, 3), P 2 (9,, 3) 7. (a) Find the terminal point of v = 3i 2j if the initial point is (, 2). (b) Find the terminal point of v = 3,,2 if the initial point is (5,0, ). 8. (a) Find the terminal point of v = 7,6 if the initial point is (2, ). (b) Find the terminal point of v = i+2j 3k if the initial point is ( 2,,4). 9 0 Perform the stated operations on the vectors u, v, and w. 9. u = 3i k, v = i j+2k, w = 3j (a) w v (b) 6u+4w (c) v 2w (d) 4(3u+v) (e) 8(v+w)+2u (f) 3w (v w) 0. u = 2,,3, v = 4,0, 2, w =,,3 (a) u w (b) 7v+3w (c) w +v (d) 3(u 7v) (e) 3v 8w (f) 2v (u+w) 2 Find the norm of v.. (a) v =, (b) v = i+7j (c) v =,2,4 (d) v = 3i+2j+k 2. (a) v = 3,4 (b) v = 2i 7j (c) v = 0, 3,0 (d) v = i+j+k 3. Let u = i 3j+2k, v = i+j, and w = 2i+2j 4k. Find (a) u+v (b) u + v (c) 2u +2 v (d) 3u 5v+w (e) w w (f) w w 4 5 Find the unit vectors that satisfy the stated conditions. 4. (a) Same direction as i+4j. (b) Oppositely directed to 6i 4j+2k. (c) Same direction as the vector from the point A(,0,2) to the point B(3,,).

4 MA2: Prepared by Dr. Archara Pacheenburawana 4 5. (a) Oppositely directed to 3i 4j. (b) Same direction as 2i j 2k. (c) Same direction as the vector from the point A( 3,2) to the point B(, ). 6 7 Find the vectors that satisfy the stated conditions. 6. (a) Oppositely directed to v = 3, 4 and half the length of v. (b) Length 7 and same direction as v = 7,0, (a) Same direction as v = 2i+3j and three times the length of v. (b) Length 2 and oppositely directed to v = 3i+4j+k. 8. In each part, find the component form of the vector v in 2-space that has the stated length and makes the stated angle θ with the positive x-axis. (a) v = 3; θ = π/4 (b) v = 2; θ = 90 (c) v = 5; θ = 20 (d) v = ; θ = π 9. Find the component formof v+w and v w in 2-space, given that v =, w =, v makes an angle of π/6 with the positive x-axis, and w makes an angle of 3π/4 with the positive x-axis. 20. Let u =,3, v = 2,, and w = 4,. Find the vector x that satisfies 2u v+ x = 7x+w. 2. Let u =,, v = 0,, and w = 3,4. Find the vector x that satisfies u 2x = x w+3v. 22. Find u and v if u+2v = 3i k and 3u v = i+j+k. 23. Find u and v if u+v = 2, 3 and 3u+2v =, In each part, find two unit vectors in 2-space that satisfy the stated condition. (a) Parallel to the line y = 3x+2 (b) Parallel to the line x+y = 4 (c) Perpendicular to the line y = 5x+ Exercise 3.3. In each part, find the dot product of the vectors and the cosine of the angle between them. (a) u = i+2j, v = 6i 8j (b) u = 7, 3, v = 0,

5 MA2: Prepared by Dr. Archara Pacheenburawana 5 (c) u = i 3j+7k, v = 8i 2j 2k (d) u = 3,,2, v = 4,2, 5 2. In each part use the given information to find u v. (a) u =, v = 2, the angle between u and v is π/6. (b) u = 2, v = 3, the angle between u and v is In each part, determine whether u and v make an acute angle, an obtuse angle, or are orthogonal. (a) u = 7i+3j+5k, v = 8i+4j+2k (b) u = 6i+j+3k, v = 4i 6k (c) u =,,, v =,0,0 (d) u = 4,,6, v = 3,0,2 4. Does the triangle in 3-space with vertices (,2,3), (2, 2,0), and (3,, 4) have an obtuse angle? Justify your answer. 5. The accompanying figure shows eight vectors that are equally spaced around a circle of radius. Find the dot product of v 0 with each of the other seven vectors. v 3 v 2 v v 4 v 0 v 5 v 6 v 7 6. The accompanying figure shows six vectors that are equally spaced around a circle of radius 5. Find the dot product of v 0 with each of the other five vectors. v 2 v v 3 v 0 v 4 v 5 7. (a) Use vectors to show that A(2,,), B(3,2, ), and C(7,0, 2) are vertices of the right triangle. At which vertex is the right angle? (b) Use vectors to find the interior angles of the triangle with vertices (,0), (2, ), and (,4).

6 MA2: Prepared by Dr. Archara Pacheenburawana 6 8. (a) Show that if v = ai+bj is a vector in 2-space, then the vectors are both orthogonal to v. v = bi+aj and v 2 = bi aj (b) Use the result in part (a) to find two unit vectors that are orthogonal to the vector v = 3i 2j. Sketch the vectors v, v, and v Explain why each of the following expressions makes no sense. (a) u (v w) (b) (u v)+w (c) u v (d) k (u+v) 0. True or false? If u v = u w and if u 0, then v = w. Justify your conclusion.. Verify part (b) and (c) of Theorem2.7 forthe vectors u = 6i j+2k, v = 2i+7j+4k, w = i+j 3k and k = Let u =,2, v = 4, 2, and w = 6,0. Find (a) u (7v+w) (b) (u w)w (c) u (v w) (d) ( u v) w 3. FindrsothatthevectorfromthepointA(,,3)tothepointB(3,0,5)isorthogonal to the vector from A to the point P(r,r,r). 4. Find two unit vectors in 2-space that make an angle of 45 with 4i+3j. 5 6 Find the direction cosines of v. 5. (a) v = i+j k (b) v = 2i 2j+k 6. (a) v = 3i 2j 6k (b) v = 3i 4k 7. In each part, find the vector component of v along b and the vector component of v orthogonal to b. (a) v = 2i j, b = 3i+4j (b) v = 4,5, b =, 2 (c) v = 3i 2j, v = 2i+j 8. In each part, find the vector component of v along b and the vector component of v orthogonal to b. (a) v = 2i j+3k, b = i+2j+2k (b) v = 4,,7, b = 2,3, 6 (c) v = 3i 2j, v = 2i+j

7 MA2: Prepared by Dr. Archara Pacheenburawana Express the vector v as the sum of a vector parallel to b and a vector orthogonal to b. 9. (a) v = 2i 4j, b = i+j (b) v = 3i+j 2k, b = 2i k (c) v = 4i 2j+6k, b = 2i+j 3k 20. (a) v = 3,5, b =, (b) v = 2,,6, b = 0, 2, (c) v =,4,, b = 3, 2,5 2. Find the work done by a force F = 3j (pounds) applied to a point that moves on the line from (,3) to (4,7), Assume that distance is measured in feet. 22. A force F = 4i 6j + k newtons is applied to a point that moves a distance of 5 meters in the direction of the vector i+j+k. How much work is done? 23. Aboat travels 00meters duenorthwhile the windthat applies a forceof500 newtons toward the northwest. How much work does the wind do? 24. A box is dragged along the floor by a rope that applies a force of 50 lb at an angle of 60 with the floor. How much work is done moving the box 5 ft? Exercise 3.4. (a) Use a determinant to find the cross product i (i+j+k) (b) Check your answer in part (a) by rewriting the cross product as and evaluate each term. i (i+j+k) = (i i)+(i j)+(i k) 2. In each part, use the two methods in Exercise to find (a) j (i+j+k) (b) k (i+j+k) 3 6 Find u v and check that it is orthogonal to both u and v. 3. u =,2, 3, v = 4,,2 4. u = 3i+2j k, v = i 3j+k 5. u = 0,, 2, v = 3,0, 4

8 MA2: Prepared by Dr. Archara Pacheenburawana 8 6. u = 4i+k, v = 2i j 7. Let u = 2,,3, v = 0,,7, and w =,4,5. Find (a) u (v w) (b) (u v) w (c) (u v) (v w) (d) (v w) (u v) 8. Find two unit vectors that are orthogonal to both u = 7i+3j+k, v = 2i+4k 9. FindtwounitvectorsthatarenormaltotheplanedeterminedbythepointsA(0, 2,), B(,, 2), and C(,,0). 0. Find two unit vectors that are parallel to the yz-plane and are orthogonal to the vector 3i j+2k. 2 Find the area of the parallelogram that has u and v as adjacent sides.. u = i j+2k, v = 3j+k 2. u = 2i+3j, v = i+2j 2k 3 4 Find the area of the triangle with vertices P, Q, and R. 3. P(,5, 2), Q(0,0,0), R(3,5,) 4. P(2,0, 3), Q(,4,5), R(7,2,9) 5 8 Find u (v w). 5. u = 2i 3j+k, v = 4i+j 3k, w = j+5k 6. u =, 2,2, v = 0,3,2, w = 4,, 3 7. u = 2,,0, v =, 3,, w = 4,0, 8. u = i, v = i+j, w = i+j+k 9 20 Use a scalar triple product to find the volume of the parallelepiped that has u, v, and w as adjacent edges. 9. u = 2, 6,2, v = 0,4, 2, w = 2,2, u = 3i+j+2k, v = 4i+5j+k, w = i+2j+4k 2. In each part, use a scalar triple product to determine whether the vectors lie in the same plane.

9 MA2: Prepared by Dr. Archara Pacheenburawana 9 (a) u =, 2,, v = 3,0, 2, w = 5, 4,0 (b) u = 5i 2j+k, v = 4i j+k, w = i j (c) u = 4, 8,, v = 2,, 2, w = 3, 4,2 22. Suppose that u (v w). Find (a) u (w v) (c) w (u v) (e) (u w) v (b) (v w) u (d) v (u w) (f) v (w w) Exercise Find parametric equations for the line through P and P 2 and also for the line segment joining those points.. (a) P (3, 2), P 2 (5,) (b) P (5, 2,), P 2 (2,4,2) 2. (a) P (0,), P 2 ( 3, 4) (b) P (,3,5), P 2 (,3,2) 3 4 Find parametric equations for the line whose vector equation is given. 3. (a) x,y = 2, 3 +t, 4 (b) xi+yj+zk = k+t(i j+k) 4. (a) xi+yj = (3i 4j)+t(2i+j) (b) x,y,z =,0,2 +t,3,0 5 6 Find a point P on the line and a vector v parallel to the line by inspection. 5. (a) xi+yj = (2i j)+t(4i j) (b) x,y,z =,2,4 +t 5,7, 8 6. (a) x,y =,5 +t 2,3 (b) xi+yj+zk = (i+j 2k)+tj 7 8 Express the given parametric equations of a line using bracket notation and also using i, j, k notation. 7. (a) x = 3+t, y = 4+5t (b) x = 2 t, y = 3+5t, z = t 8. (a) x = t, y = 2+t (b) x = +t, y = 7+3t, z = 4 5t

10 MA2: Prepared by Dr. Archara Pacheenburawana Find parametric equations of the line that satisfies that stated conditions. 9. The line through ( 5,2) that is parallel to 2i 3j. 0. The line through (0,3) that is parallel to the line x = 5+t, y = 2t.. The line that is tangent to the circle x 2 +y 2 = 25 at the point (3, 4). 2. The line that is tangent to the parabola y = x 2 at the point ( 2,4). 3. The line through (,2,4) that is parallel to 3i 4j+k. 4. The line through (2,,5) that is parallel to,2,7. 5. The line through ( 2,0,5) that is parallel to the line x = +2t, y = 4 t, z = 6+2t. 6. The line through the origin that is parallel to the line x = t, y = +t, z = Where does the line x = +3t, y = 2 t intersect (a) the x-axis (c) the parabola y = x 2? (b) the y-axis 8. Where does the line x,y = 4t,3t intersect the circle x 2 +y 2 = 25? 9 20 Find the intersections of the lines with xy-plane, the xz-plane, and the yz-plane. 9. x = 2, y = 4+2t, z = 3+t 20. x = 2t, y = 3+t, z = 4 t 2. Where does the line x = +t, y = 3 t, z = 2t intersect the cylinder x 2 +y 2 = 6? 22. Where does the line x = 2 t, y = 3t, z = +2t intersect the plane 2y +3z = 6? Show that the line L and L 2 intersect, and find their point of intersection. 23. L : x = 2+t, y = 2+3t, z = 3+t L 2 : x = 2+t, y = 3+4t, z = 4+2t 24. L : x+ = 4t, y 3 = t, z = 0 L 2 : x+3 = 2t, y = 6t, z 2 = 3t Show that the line L and L 2 are skew. 25. L : x = +7t, y = 3+t, z = 5 3t L 2 : x = 4 t, y = 6, z = 7+2t

11 MA2: Prepared by Dr. Archara Pacheenburawana 26. L : x = 2+8t, y = 6 8t, z = 0t L 2 : x = 3+8t, y = 5 3t, z = 6+t Determine whether the line L and L 2 are parallel. 27. L : x = 3 2t, y = 4+t, z = 6 t L 2 : x = 5 4t, y = 2+2t, z = 7 2t 28. L : x = 5+3t, y = 4 2t, z = 2+3t L 2 : x = +9t, y = 5 6t, z = 3+8t Determine whether the point P, P 2, and P 3 lie on the same line. 29. P (6,9,7), P 2 (9,2,0), P 3 (0, 5, 3) 30. P (,0,), P 2 (3, 4, 3), P 3 (4, 6, 5) Exercise Find an equation of the plane that passes through the point P and has the vector n as normal.. P(2,6,); n =,4,2 2. P(,,2); n =,7,6 3. P(,0,0); n = 0,0, 4. P(0,0,0); n = 2, 3, Find an equation of the plane indicated in the figure z z 5. y 6. y x z x z 7. y 8. y x x

12 MA2: Prepared by Dr. Archara Pacheenburawana Find an equation of the plane that passes through the given point. 9. ( 2,,), (0,2,3), and (,0, ) 0. (3,2,), (2,, ), and (,3,2) 2 Determine whether the planes are parallel, perpendicular, or neither. 3. (a) 2x 8y 6z 2 = 0 x+4y +3z 5 = 0 (c) x y +3z 2 = 0 2x+z = 4. (a) 3x 2y +z = 4 6x 4y +3z = 7 (b) 3x 2y +z = 4x+5y 2z = 4 (b) y = 4x 2z +3 x = 4 y + 2 z (c) x+4y +7z = 3 5x 3y +z = Determine whether the line and planes are parallel, perpendicular, or neither. 3. (a) x = 4+2t, y = t, z = 4t; 3x+2y +z 7 = 0 (b) x = t, y = 2t, z = 3t; x y +2z = 5 (c) x = +2t, y = 4+t, z = t; 4x+2y 2z = 7 4. (a) x = 3 t, y = 2+t, z = 3t; 2x+2y 5 = 0 (b) x = 2t, y = t, z = t; 6x 3y +3z = (c) x = t, y = t, z = 2+t; x+y +z = 5 6 Determine whether the line and planes intersect; if so, find the coordinates of the intersection.

13 MA2: Prepared by Dr. Archara Pacheenburawana 3 5. (a) x = t, y = t, z = t; 3x 2y +z 57 = 0 (b) x = 2 t, y = 3+t, z = t; 2x+y +z = 6. (a) x = 3t, y = 5t, z = t; 2x y +z + = 0 (b) x = +t, y = +3t, z = 2+4t; x y +4z = Find the acute angle of intersection of the planes. 7. x = 0 and 2x y +z 4 = 0 8. x+2y 2z = 5 and 6x 3y +2z = Find an equation of the plane that satisfies the stated conditions. 9. The plane through the origin that is parallel to the plane 4x 2y +7z +2 = The plane that contains the line x = 2+3t, y = 4+2t, z = 3 t and is perpendicular to the plane x 2y +z = The plane through the point (,4,2) that contains the line of intersection of the planes 4x y +z 2 = 0 and 2x+y 2z 3 = The plane through (,4, 3) that is perpendicular to the line x 2 = t, y+3 = 2t, and z = t. 23. The plane through (,2, ) that is perpendicular to the line of intersection of the planes 2x+y +z = 2 and x+2y +z = The plane through the points P ( 2,,4), P 2 (,0,3) that is perpendicular to the planes 4x y +3z = The plane through (,2, 5) that is perpendicular to the planes 2x y+z = and x+y 2z = The plane that contains the point (2, 0, 3) and the line x = + t, y = t, and z = 4+2t. 27. The plane whose points are equidistant from (2,,) and (3,,5). 28. The plane that contains the line x = 3t, y = + t, z = 2t and is parallel to the intersection of the planes y +z = and 2x y +z = 0.

14 MA2: Prepared by Dr. Archara Pacheenburawana Find parametric equations of the line through the point (5,0, 2) that is parallel to the planes x 4y +2z = 0 and 2x+3y z + = Let L be the line x = 3t+, y = 5t, z = t. (a) Show that L lies in the plane 2x+y z = 2. (b) Show that L is parallel to the plane x+y +2z = 0. Is the line above, below, or on this plane? 3 32 Find the distance between the point and the plane. 3. (, 2,3); 2x 2y +z = (0,,5); 3x+6y 2z 5 = Find the distance between parallel planes. 33. (a) 2x+y +z = 0 6x 3y 3z 5 = (b) x+y +z = x+y +z = Find the distance between the given shew lines. 35. x = +7t, y = 3+t, z = 5 3t x = 4 t, y = 6, z = 7+2t 36. x = 3 t, y = 4+4t, z = +2t x = t, y = 3, z = 2t 37. Find an equation of the sphere with center (2,, 3) that is tangent to the plane x 3y +2z = Locate the point of intersection of the plane 2x + y z = 0 and the line through (3,,0) that is perpendicular to the plane. Exercise 3.7. Identify the quadric surface as an ellipsoids, hyperboloids of one sheet, hyperboloids of two sheet, elliptic cones, elliptic paraboloids, and hyperbolic paraboloids. State the value of a, b, and c in each case. (a) z = x2 4 + y2 9 (b) z = y2 25 x2 (c) x 2 +y 2 z 2 = 6 (d) x 2 +y 2 z 2 = 0 (e) 4z = x 2 +4y 2 (f) z 2 x 2 y 2 =

15 MA2: Prepared by Dr. Archara Pacheenburawana 5 2. Find an equation of the trace, and state whether it is an ellipse, a parabola, or a hyperbola (a) 4x 2 +y 2 +z 2 = 4; y = (b) 4x 2 +y 2 +z 2 = 4; x = 2 (c) 9x 2 y 2 z 2 = 6; x = 2 (d) 9x 2 y 2 z 2 = 6; z = 2 (e) z = 9x 2 +4y 2 ; y = 2 (f) z = 9x 2 +4y 2 ; z = Identify and sketch the quadric surface. 3. x 2 + y2 4 + z2 9 = 4. x2 4 + y2 9 z2 6 = 5. 4z 2 = x 2 +4y z 2 4y 2 9x 2 = z = y 2 x z = x 2 +2y 2 Exercise Convert from rectangular to cylindrical coordinates.. (4 3,4, 4) 2. ( 5,5,6) 3. (0,2,0) 4. (4, 4 3,6) 5 8 Convert from cylindrical to rectangular coordinates. 5. (4,π/6, 2) 6. (8,3π/4, 2) 7. (5,0,4) 8. (7,π, 9) 9 2 Convert from rectangular to spherical coordinates. 9. (, 3, 2) 0. (,, 2). (0,3 3,3) 2. ( 5 3,5,0) 3 6 Convert from spherical to rectangular coordinates. 3. (5,π/6,π/4) 4. (7,0,π/2) 5. (,π,0) 6. (2,3π/2,π/2) 7 20 Convert from cylindrical to spherical coordinates.

16 MA2: Prepared by Dr. Archara Pacheenburawana 6 7. ( 3,π/6,3) 8. (,π/4, ) 9. (2,3π/4,0) 20. (6,, 2 3) 2 24 Convert from spherical to cylindrical coordinates. 2. (5,π/4,2π/3) 22. (,7π/6,π) 23. (3,0,0) 24. (4,π/6,π/2) An equation is given in cylindrical coordinates. Express the equation in rectangular coordinates. 25. r = z = r r = 4sinθ 28. r 2 +z 2 = An equation is given in spherical coordinates. Express the equation in rectangular coordinates. 29. ρ = φ = π/4 3. ρ = 4cosφ 32. ρsinφ = 2cosθ An equation of a surface is given in rectangular coordinates. Find an equation of a surface in (a) cylindrical coordinates and (b) spherical coordinates. 33. z = z = 3x 3 +3y x 2 +y 2 = x 2 +y 2 +z 2 = x+3y +4z = 38. x 2 = 6 z 2 Extra Problems. If a bug walks on the sphere x 2 +y 2 +z 2 +2x 2y 4z 3 = 0 how close and how far can it get from the origin? 2. Write the equations of the following two spheres: Sphere A: center (2, 3,4) and radian 3, Sphere B: center (4,3, 5) and radian 4. (a) What is the minimum distance between a point on A and a point on B?

17 MA2: Prepared by Dr. Archara Pacheenburawana 7 (b) What is the maximum distance between a point on A and a point on B? 3. What is the equation of the sphere with center (,2,3) which passes through the point (0,4,)? 4. Find the equation of the sphere with center (,2,4) that touches the xz-plane. 5. Two vectors u and v are of the form u = a( i+j) and v = bi+j where a and b are scalars. If u+v+0i j = 0, find the scalars a and b. 6. Find the vector component of v = 2i j+3k along b = i+2j+2k and the vector component of v orthogonal to b. 7. Find a vector a such that the norm of the projection of a on b is 2, where b = 4, 2, 4. (There are several a satisfying the condition above.) 8. A force F = 4i 6j + k newtons is applied to a point that moves a distance of 5 meters in the direction of the vector i+j+k. How much work is done? 9. Given two vectors u = 3,2,2 and v = 4,3,. (a) Find a unit vector in the same direction as u. (b) Find the angle between u and v. (c) Find the direction cosines of u. (d) Find the orthogonal projection of u on v. 0. A tow truck drags a stalled car along a road. The chain makes an angle of 30 or π/6 radians with the road. The tension (force) in the chain is 500 N. How much work is done by the truck in pulling the car km; i.e., 000 m?. A sport utility vehicle with a gross weight of 5400 pounds is parked on a slope of 30. Assume that the only force the only force to overcome is the force of gravity. (a) Find the force required to keep the vehicle from rolling down the hill. (b) Find the force perpendicular to the hill. 2. Three of the four vertices of a parallelogram are P(0,,), Q(0,,0), and R(3,,). Two of the sides are PQ and PR. (a) Find the area of the parallelogram. (b) Find the cosine of the angle between the vectors PQ and PR. (c) Find the orthogonal projection of PQ on PR. 3. Given points P(,0,0),Q(0,2,0), and R(0,0,3) in 3-space. (a) Find a vector orthogonal to the plane passing through P,Q, and R.

18 MA2: Prepared by Dr. Archara Pacheenburawana 8 (b) Find the area of triangle PQR. 4. Use a scalar triple product to find the volume of the parallelepiped that has u = 3i+j+2k, v = 4i+5j+k, and w = i+2j+4k as adjacent edges. 5. Find the volume of the parallelepiped with three adjacent edges formed by u =,0,2, v = 0,2,3, and w = 0,,3. 6. Let a and b be two vectors such that a b = 2i+2j 3k. (a) What is ( 3b) (4a)? (b) Find a (a b) if it exists. If it doesn t exist then explain why not. (c) Find a (a b) if it exists. If it doesn t exist then explain why not. (d) Suppose that not only does a b = 2i+2j 3k, but also that a b = a b. Which of the following statements must be true? i. a b ii. a and b are parallel iii. a is perpendicular to b iv. either a or b must be a unit vector 7. Find parametric equations of the line L passing through the points P(0,2,) and Q(2,0,2). 8. FindanequationoftheplanethroughthepointsP ( 2,,),P 2 (0,2,3),andP 3 (,0, ). 9. Given a point P(0,,2) and the vectors u =,0, and v = 2,3,0, find (a) an equation for the plane that contains P and whose normal vector is perpendicular to the two vectors u and v. (b) Find the distance from the point (2,, ) to the plane from part (a). 20. Find the distance between parallel planes 2x+y +z = 0 and 6x 3y 3z 5 = 0 2. Find parametric equations of the line L that passed through the point ( 2,0,5) and parallel to the line x = +2t, y = 4 t, z = 6+2t. 22. Are the lines L and M given below parallel or skew or intersect? Explain your answer. If they intersect, find the intersection point. where s and t are parameters. L : x = 2t, y = 2t, z = 5 t M : x = 3+2s, y = 2, z = 3+2s

19 MA2: Prepared by Dr. Archara Pacheenburawana Let L and L 2 be the lines L : x = +2t, y = 3t, z = 2 t L 2 : x = +t, y = 4+t, z = +3t Determine whether the lines L and L 2 given above are parallel, skew or intersecting. 24. (a) Find an equation of the plane containing the lines x = 4 4t, y = 3 t, z = +5t and x = 4 t, y = 3+2t, z = (b) Find the distance from the point (,0, ) to the plane 2x+y 2z =. (c) FindthepointP intheplane2x+y 2z = which isclosest tothepoint (,0, ). (Hint: You can use part (b) of this problem to help find P.) 25. Let x z = be the equation of the plane P, y + 2z = 3 be the equation of the plane P 2, and x+y 2z = be the equation of the plane P 3. Let L be the line of intersection of the planes P and P 2. (a) What is the normal vector to the plane P? (b) Find the acute angle of intersection between the planes P and P 2. (c) Find the parametric equations of the line L. (d) Find the equation of the plane P passes through the line L and is perpendicular to the plane P Find the spherical coordinates of the point that has rectangular coordinates ( (x,y,z) =,, ) 2 ( 2,π, π ) 8. Given as a point in spherical coordinates. 2 ( 2,π, π ) (a) The rectangular coordinates of are... 2 ( 2,π, π ) (b) The cylindrical coordinates of are (a) Find the spherical coordinates of the point that has rectangular coordinates (,, 2 ). (b) An equation of a surface x 2 = 6 z 2 is given in rectangular coordinates. Find an equation of the surface in cylindrical coordinates. 28. The equation of the hyperboloid of one sheet is of the kind: x 2 +y 2 = +z 2. Rewrite this equation using the cylindrical and spherical coordinate systems.

20 MA2: Prepared by Dr. Archara Pacheenburawana For the surface x 2 +4y 2 +z 2 2x = n. (a) If n = 0, what quadric surface is this? Explain. (b) If n > 0, what quadric surface is this? Explain.