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All Textbook Solutions for Calculus (MindTap Course List)
What does the equation x=4 represent in 2? What does it represent in 3? Illustrate with sketches.What does the equation y = 3 represent in 3? What does z = 5 represent? What does the pair of equations y = 3, z = 5 represent? In other words, describe the set of points x, y, z such that y = 3 and z = 5. Illustrate with a sketch.Describe and sketch the surface in 3 represented by the equation x+y=2.8EFind the lengths of the sides of the triangle PQR. Is it a right triangle? Is it an isosceles triangle? P(3,2,3),Q(7,0,1),R(1,2,1)Find the lengths of the sides of the triangle PQR. Is it a right triangle? Is it an isosceles triangle? P(2,1,0),Q(4,1,1),R(4,5,4)Determine whether the points lie on a straight line. a A(2,4,2),B(3,7,2),C(1,3,3) b D(0,5,5),E(1,2,4),F(3,4,2)12EFind an equation of the sphere with centre (3,2,5) and radius 4. What is the intersection of this sphere with the yz-plane?Find an equation of the sphere with centre (2,6,4) and radius 5. Describe its intersection with each of the coordinate planes.Find an equation of the sphere that passes through the point (4,3,1) and has centre (3,8,1).16E17E18E19E20Ea Prove that the midpoint of the line segment from P1(x1,y1,z1) to P2(x2,y2,z2) is (x1+x22,y1+y22,z1+z22) b Find the lengths of the medians of the triangle with vertices A(1,2,3),B(2,0,5), and C(4,1,5). A median of a triangle is a line segment that joins a vertex to the midpoint of the opposite side.Find an equation of a sphere if one of its diameters has endpoints (5,4,3) and (1,6,9).23EFind an equation of the largest sphere with centre (5,4,9) that is contained in the first octant.25EDescribe in words the region of 3 represented by the equations or inequality. y=227E28EDescribe in words the region of 3 represented by the equations or inequality. 0z630EDescribe in words the region of 3 represented by the equations or inequality. x2+y2=4,z=1Describe in words the region of 3 represented by the equations or inequality. x2+y2=433EDescribe in words the region of 3 represented by the equations or inequality. x2+y2+z24Describe in words the region of 3 represented by the equations or inequality. 1x2+y2+z2536EDescribe in words the region of 3 represented by the equations or inequality. x2+y2938E39EWrite inequalities to describe the region. The solid cylinder that lies on or below the plane z=8 and on or above the disk in the xy-plane with centre the origin and radius 2Write inequalities to describe the region. The region consisting of all points between but not on the spheres of radius r and R centred at the origin, where rRWrite inequalities to describe the region. The solid upper hemisphere of the sphere of radius 2 centred at the originThe figure shows a line L1 in space and a second line L2, which is the projection of L1 onto the xy-plane. In other words, the points on L2 are directly beneath, or above, the points on L1. a Find the coordinates of the point P on the line L1. b Locate on the diagram the points A, B, and C, where the line L1 intersects the xy-plane, the yz-plane, and the xz-plane, respectively.Consider the points P such that the distance from P to A(1,5,3) is twice the distance from P to B(6,2,2). Show that the set of all such points is a sphere, and find its center and radius.45EFind the volume of the solid that lies inside both of the spheres x2+y2+z2+4x2y+4z+5=0 and x2+y2+z2=4Find the distance between the spheres x2+y2+z2=4 and x2+y2+z2=4x+4y+4z11Describe and sketch a solid with the following properties. When illuminated by rays parallel to the z-axis, its shadow is a circular disk. If the rays are parallel to the y-axis, its shadow is a square. If the rays are parallel to the x-axis, its shadow is an isosceles triangle.Are the following quantities vectors or scalars? Explain. a The cost of a theater ticket b The current in a river c The initial flight path from Houstan to Dallas d The population of the worldWhat is the relationship between the point 4, 7 and the vector 4,7? Illustrate with a sketch.Name all the equal vectors in the parallelogram shown.4E5E6EIn the figure, the tip of c and the tail of d are both the midpoint of QR. Express c and d in terms of a and b.If the vectors in the figure satisfy |u|=|v|=1 and u+v+w=0, what is |w|?Find a vector a with representation given by the directed line segment AB. Draw AB and the equivalent representation starting at the origin. A(2,1),B(1,2)10E11E12E13E14E15E16E17E18E19EFind a+b,4a+2b,|a|, and |ab|. a=5i+3j,b=i2j21EFind a+b,4a+2b,|a|, and |ab|. a=8,1,4,b=5,2,123E24E25E26EWhat is the angle between the given vector and the positive direction of the x-axis? i+3j28EIf v lies in the first quadrant and makes an angle /3 with the positive x-axis and |v|=4, find v in component form.30E31EFind the magnitude of the resultant force and the angle it makes with the positive x-axis.Find the magnitude of the resultant force and the angle it makes with the positive x-axis.34E35E36EA block-and-tackle pulley hoist is suspended in a warehouse by ropes of lengths 2 m and 3 m. The hoist weighs 350 N. The ropes, fastened at different heights, make angles of 50 and 38 with the horizontal. Find the tension in each rope and the magnitude of each tension.The tension T at each end of a chain has magnitude 25 N see the figure. What is the weight of the chain?A boatman wants to cross a canal that is 3 km wide and wants to land at a point 2 km upstream from his starting point. The current in the canal flows at 3.5 km/h and the speed of his boat is 13 km/h. a In what direction should he steer? b How long will the trip take?Three forces act on an object. Two of the forces are at an angle of 100 to each other and have magnitudes 25 N and 12 N. The third is perpendicular to the plane of these two forces and has magnitude 4 N. Calculate the magnitude of the force that would exactly counterbalance these three forces.41E42EIf A, B, and C are the vertices of a triangle, find AB+BC+CA44Ea Draw the vectors a=3,2,b=2,1 and c=7,1. b Show, by means of a sketch, that there are scalars s and t such that c=sa+tb. c Use the sketch to estimate the values of s and t. d Find the exact values of s and t.46E47E48E49EProve Property 5 of vectors algebraically for the case n = 3. Then use similar triangles to give a geometric proof.Use vectors to prove that the line joining the midpoints of two sides of a triangle is parallel to the third side and half its length.52EWhich of the following expressions are meaningful? Which are meaningless ?Explain. a (ab)c b (ab)c c |a|(bc) d a(b+c) e ab+c f |a|(b+c)2E210 Find ab. a=1.5,0.4, b=4,64E5E6E7E8E9E10EIf u is a unit vector, find uv and uw.If u is a unit vector, find uv and uw.13E14E15EFind the angle between the vectors. First find an exact expression and then approximate to the nearest degree. a=2,5,b=5,1217E18E19E20E2122 Find, correct to the nearest degree, the three angles of the triangle with the given vertices. P(2,0),Q(0,3),R(3,4)22E23E2324 Determine whether the given vectors are orthogonal, parallel, or neither a u=5,4,2,v=3,4,1 b u=9i6j+3k,v=6i+4j2k c u=c,c,c,v=c,0,c25E26E27E28E29E30E3132 Find the acute angles between the curves at their points of intersection. The angle between two curves is the angle between their tangent lines at the point of intersection. y=x2, y=x33132 Find the acute angles between the curves at their points of intersection. The angle between two curves is the angle between their tangent lines at the point of intersection. y=sinx, y=cosx, 0x/233E34E35E36E37E38E3944 Find the scalar and vector projections of b onto a. a=5,12, b=4,640E41EFind the scalar and vector projections of b onto a. a=1,4,8, b=12,1,2Find the scalar and vector projections of b onto a. a=3i3j+k, b=2i+4jk44E45E46EIf a=3,0,1, find a vector b such that compab=2.48E49E50E51E52EUse a scalar projection to show that the distance from a point P1(x1,y1) to the line ax+by+c=0 is |ax1+by1+c|a2+b2 Use this formula to find the distance from the point (2,3) to the line to the line 3x4y+5=0.If r=x,y,z,a=a1,a2,a3, and b=b1,b2,b3, show that the vector equation (r-a)(r-b)=0 represents a sphere, and find its center and radius.55E56E57EIf c=|a|b+|b|a, where a, b, and c are all nonzero vectors, show that c bisects the angle between a and b.59E60EUse Theorem 3 to prove the Cauchy-Schwarz Inequality: |ab||a||b|62EThe Parallelogram Law states that |a+b|2+|ab|2=2|a|2+2|b|2 aGive a geometric interpretation of the Parallelogram Law. bProve the Parallelogram Law. See the hint in Exercise 62.64E65E1E2E17 Find the cross product ab and verify that it is orthogonal to both a and b. a=2j4k, b=i+3j+k17 Find the cross product ab and verify that it is orthogonal to both a and b. a=3i+3j3k, b=3i3j+3k5E6E7EIf a=i2k and b=j+k, find ab. Sketch a, b, and ab. as vectors starting at the origin.9E912 Find the vector, not with determinants, but by using properties of cross products. k(i2j)912 Find the vector, not with determinants, but by using properties of cross products. (jk)(ki)912 Find the vector, not with determinants, but by using properties of cross products. (i+j)(ij)13E14E15E16E17EIf a=1,0,1,b=2,1,1, and c=0,1,3, show that a(bc)(ab)cFind two unit vectors orthogonal to both 3,2,1 and 1,1,0.20E21E22E23E24E25E26E27EFind the area of the parallelogram with vertices P(1,0,2),Q(3,3,3),R(7,5,8), and S(5,2,7).2932 a Find a nonzero vector orthogonal to the plane through the points P, Q, and R, and b find the area of triangle PQR. P(1,0,1),Q(2,1,3),R(4,2,5)2932 a Find a nonzero vector orthogonal to the plane through the points P, Q, and R, and b find the area of triangle PQR. P(0,0,3),Q(4,2,0),R(3,3,1)2932 a Find a nonzero vector orthogonal to the plane through the points P, Q, and R, and b find the area of triangle PQR. P(0,2,0),Q(4,1,2),R(5,3,1)32E33E34E35E36E37E38E39E40EA wrench 30 cm long lies along the positive y-axis and grips a bolt at the origin. A force is applied in the direction 0,3,4 at the end of the wrench. Find the magnitude of the force needed to supply 100 Nm of torque to the bolt.42E43Ea Find all vectors v such that 1,2,1v=3,1,5 b Explain why there is no vector v such that 1,2,1v=3,1,545Ea Let P be a point not on the plane that passes through the points Q, R, and S. Show that the distance d from P to the plane is d=|a(bc)||ab| where a=QR, b=QS, and c=QP. b Use the formula in part a to find the distance from the point P(2,1,4) to the plane through the points Q(1,0,0), R(0,2,0), and S(0,0,3).47E48E49E50E51E52ESuppose that a0. a If ab=ac, does it follow that b=c? b If ab=ac, does it follow that b=c? c If ab=ac, and ab=ac, does it follow that b=c?If v1, v2, and v3 are noncoplanar vectors, let k1=v2v3v1(v2v3)k2=v3v1v1(v2v3) k3=v1v2v1(v2v3) These vectors occur in the study of crystallography. Vectors of the from n1v1+n2v2+n3v3, where each ni is an integer, from a lattice for a crystal. Vectors written similarly in terms of k1,k2, and k3 from the reciprocal lattice. a Show that ki is perpendicular to vj if ij. b Show that kivi=1 for i=1,2,3. c Show that k1(k2k3)=1v1(v2v3).Determine whether each statement is true or false in 3. a Two lines parallel to a third line are parallel. b Two lines perpendicular to a third line are parallel. c Two planes parallel to a third plane are parallel. d Two planes perpendicular to a third plane are parallel. e Two lines parallel to a plane are parallel. f Two lines perpendicular to a plane are parallel. g Two planes parallel to a line are parallel. h Two planes perpendicular to a line are parallel. i Two planes either intersect or are parallel. j Two lines either intersect or are parallel. k A plane and a line either intersect or are parallel.2E3E4E5E612 Find parametric equations and symmetric equations for the line. The line through the origin and the point (4,3,1)7E8E9E612 Find parametric equations and symmetric equations for the line. The line through (2,1,0) and perpendicular to both i+j and j+k612 Find parametric equations and symmetric equations for the line. The line through (6,2,3) and parallel to the line 12x=13y=z+1612 Find parametric equations and symmetric equations for the line. The line of intersection of the planes x+2y+3z=1 and xy+z=113EIs the line through (2,4,0) and (1,1,1) perpendicular to the line through (2,3,4) and (3,1,8)?a Find symmetric equations for the line that passes through the point (1,5,6) and is parallel to the vector 1,2,3. b Find the points in which the required line in part a intersects the coordinate planes.a Find parametric equations for the line through 2, 4, 6 that is perpendicular to the plane xy+3z=7. b In what points does this line intersect the coordinate planes?17E18E1922 Determine whether the lines L1 and L2 are parallel, skew, or intersecting. If they intersect, find the point of intersection. L1:x=3+2t, y=4t, z=1+3t L2:x=1+4s, y=32s, z=4+5s20E21E22E2340 Find an equation of the plane. The plane through the origin and perpendicular to the vector 1,2,52340 Find an equation of the plane. The plane through the point (5,3,5) and with normal vector 2i+jk2340 Find an equation of the plane. The plane through the point (1,12,3) and with normal vector i+4j+k2340 Find an equation of the plane. The plane through the point (2,0,1) and perpendicular to the line x=3t,y=2t,z=3+4t2340 Find an equation of the plane. The plane through the point (1,1,1) and parallel to the plane 5xyz=62340 Find an equation of the plane. The plane through the point (3,2,8) and parallel to the plane z=x+y2340 Find an equation of the plane. The plane through the point (1,12,13) and parallel to the plane x+y+z=02340 Find an equation of the plane. The plane that contains the line x=1+t,y=2t,z=43t and is parallel to the plane 5x+2y+z=12340 Find an equation of the plane. The plane through the points (0,1,1), (1,0,1), and (1,1,0)2340 Find an equation of the plane. The plane through the origin and the points (3,2,1) and (1,1,1)2340 Find an equation of the plane. The plane through the points (2,1,2), (3,8,6), and (2,3,1)2340 Find an equation of the plane. The plane through the points (3,0,1), (2,2,3), and (7,1,4)2340 Find an equation of the plane. The plane that passes through the point (3,5,1) and contains the line x=4t, y=2t1, z=3t2340 Find an equation of the plane. The plane that passes through the point (6,1,3) and contains the line with symmetric equations x/3=y+4=z/22340 Find an equation of the plane. The plane that passes through the point (3,1,4) and contains the line of intersection of the planes x+2y+3z=1 and 2xy+z=32340 Find an equation of the plane. The plane that passes through the points (0,2,5) and (1,3,1) and is perpendicular to the plane 2z=5x+4y2340 Find an equation of the plane. The plane that passes through the point (1,5,1) and is perpendicular to the planes 2x+y2z=2 and x+3z=440E41E42E43E44E4547 Find the point at which the line intersects the given plane. x=22t, y=3t, z=1+t; x+2yz=746E47E48EFind direction numbers for the line of intersection of the planes x+y+z=1 and x+z=0.50E51E52E5156 Determine whether the planes are parallel, perpendicular, or neither. If neither, find the angle between them. Round to one decimal place. x+2yz=2, 2x2y+z=15156 Determine whether the planes are parallel, perpendicular, or neither. If neither, find the angle between them. Round to one decimal place. xy+3z=1, 3x+yz=25156 Determine whether the planes are parallel, perpendicular, or neither. If neither, find the angle between them. Round to one decimal place. 2x3y=z, 4x=3+6y+2z5156 Determine whether the planes are parallel, perpendicular, or neither. If neither, find the angle between them. Round to one decimal place. 5x+2y+3z=2, y=4x6z57E58E59E5960 Find symmetric equations for the line of intersection of the planes. z=2xy5, z=4x+3y561E62E63E64E65E66E67E68E69E70E7172 Find the distance from the point to the given plane. (1,2,4), 3x+2y+6z=572E73E74E75E76EShow that the lines with symmetric equations x=y=z and x+1=y/2=z/3 are skew, and find the distance between these lines.78E79E80E81E82EIf a, b, and c are not all 0, show that the equation ax+by+cz+d=0 represents a plane and a,b,c is a normal vector to the plane. Hint: Suppose a0 and rewrite the equation in the form a(x+da)+b(y0)+c(z0)=0a What does the equation y=x2 represent as a curve in 2? b What does it represent as a surface in 3? c What does the equation z=y2 represent?a Sketch the graph of y=ex as a curve in 2. b Sketch the graph of y=ex as a surface in 3. c Describe and sketch the surface z=ey.