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All Textbook Solutions for Multivariable Calculus
7E8E9E10EFinding the Angle Between Two Vectors In Exercises 1118, find the angle between the vectors (a) in radians and (b) in degrees. u= 1,1 ,v= 2,212EFinding the Angle Between Two Vectors In Exercises 1118, find the angle between the vectors (a) in radians and (b) in degrees. u=3i+j,v=2i+4j14E15E16E17E18EAlternative Form of Dot Product In Exercises 19 and 20, use the alternative form of the dot product to find uv. u=8,v=5, and the angle between u and v is /3.20E21E22E23E24E25E26EClassifying a TriangleIn Exercises 2730, the vertices of triangle are given. Determine whether the triangle is an acute triangle, an obtuse triangle, or a right triangle. Explain your reasoning. (1,2,0),(0,0,0),(2,1,0)Classifying a TriangleIn Exercises 2730, the vertices of triangle are given. Determine whether the triangle is an acute triangle, an obtuse triangle, or a right triangle. Explain your reasoning. (3,0,0),(0,0,0),(1,2,3)Classifying a TriangleIn Exercises 2730, the vertices of triangle are given. Determine whether the triangle is an acute triangle, an obtuse triangle, or a right triangle. Explain your reasoning. (2,0,1),(0,1,2),(0.5,1.5,0)30E31E32E33E34E35E36E37E38E39EFinding the Projection of u onto v In Exercises 37-44, (a) find the projection of u onto v and (b) find the vector component of u orthogonal to v. u=2i3j,v=3i+2j41E42EFinding the Projection of u onto v In Exercises 37-44, (a) find the projection of u onto v and (b) find the vector component of u orthogonal to v. u=9i2j4k,v=4j+4k44E45EProjection What can be said about the vectors u and v when projection of u onto v equals u?47E48E49ERevenueRepeat Exercises 49 after decreasing the prices by 2%. Identify the vector operation used to decrease the prices by 2%.51E52E53E54E55E56E57E58E59E60E61E62E63E64E65E66E67E68E69E70EBond AngleConsider a regular tetrahedron with vertices (0,0,0),(k,k,0),(k,0,k) and (0,k,k), where k is a positive real number. (a) Sketch the graph of the tetrahedron. (b) Find the length of each edge. (c) Find the angle between any two edges. (d) Find the angle between the line segments from the centroid (k/2,k/2,k/2) to two vertices. This is the bond angle for a molecule, such as CH4 (methane) or PbCl4 (lead tetrachloride), where the structure of the molecule is a tetrahedron.72E73E74E75ECONCEPT CHECK Vectors Explain what uv represents geometrically.CONCEPT CHECK Area Explain how to find the area of a parallelogram using vectors.3ECross Product of Unit VectorsIn Exercises 36, find the cross product of the unit vectors and sketch your result. jkCross Product of Unit Vectors In Exercises 3-6, find the cross product of the unit vectors and sketch your result. ik6E7E8EFinding Cross Products in Exercises 7-10, find (a) uv, (b) vu, and (c) vv u=7,3,2v=1,1,510E11E12E13E14E15E16E17E18E19E20E21E22E23E24E25E26ETorque The brakes on a bicycle are applied using a downward force of 20 pounds on the pedal when the crank makes a 40 angle with the horizontal (sec figure). The crank is 6 inches in length. Find the torque at P.28E29E30EFinding a Triple Scalar Product In Exercises 31-34. find u(vw). u=jv=jw=k32E33E34EVolume In Exercises 35 and 36, use t triple scalar product to find the volume of t parallelepiped having adjacent edges u, v, and w. u=i+jv=j+kw=i+k36EVolume In Exercises 37 and 38, find the volume of the parallelepiped with the given vertices. (0,0,0),(3,0,0),(0,5,1),(2,0,5),(3,5,1),(5,0,5),(2,5,6),(5,5,6)38EEXPLORING CONCEPTS Comparing Dot Products Identify the dot products that are equal. Explain your reasoning. (Assume u, v, and w are nonzero vectors.) (a) u(vw) (c) (uv)w (e) u(wv) (g) (uv)w (b) (vw)u (d) (uw)v (f) w(vu) (h) (wu)v40EEXPLORING CONCEPTS Cross ProductTwo nonzero vectors lie in the yz-plane. Where does the cross product of the vectors lie? Explain.42E43E44E45E46E47E48E49E50E51E52E53EProof Prove that u(vw)=(uw)v(uv)w.55ECONCEPT CHECK Parametric and Symmetric EquationsGive the parametric equations and the symmetric equations of a line in space. Describe what is required to find these equations.2E3E4EChecking Points on a Line In Exercises 5 and 6, determine whether each point lies on the line. x=2+t,y=3t,z=4+t (a) (0,6,6) (b) (2,3,5) (c) (4,6,2)6E7E8E9E10E11EFinding Parametric and Symmetric EquationsIn Exercises 712, find sets of (a) parametric equations and (b) symmetric equations of the line that passes through the given point and is parallel to the given vector or line. (For each line, write the direction numbers as integers.) PointParallel to (3,5,4) x13=Y+12=Z3Finding Parametric and Symmetric EquationsIn Exercises 1316, find sets of (a) parametric equations and (b) symmetric equations of the line that passes through the two points (if possible). (For each line, write the direction numbers as integers.) (5,3,2),(23,23,1)14E15EFinding Parametric and Symmetric Equations In Exercises 13-16, find sets of (a) parametric equations and (b) symmetric equations of the line that passes through the two points (if possible). (For each line, write the direction numbers as integers.) (0,0,25),(10,10,0)17E18E19E20E21E22E23E24E25E26E27EUsing Parametric and Symmetric EquationsIn Exercises 2528, find the coordinates of a point P on the line and a vector v parallel to the line. x+35=y8=z3629E30E31E32EFinding a Point of IntersectionIn Exercises 3336, determine whether the lines intersect, and if so, find the point of intersection and the angle between the lines. x=4t+2,y=3,z=t+1x=2s+2,y=2s+3,z=s+134EFinding a Point of IntersectionIn Exercises 3336, determine whether the lines intersect, and if so, find the point of intersection and the angle between the lines. x3=y21=z+1,x14=y+2=z+3336E37EChecking Points in a Plane In Exercises 37 and 38, determine whether each point lies in the plane. 2x+y+3z6=0 (a) (3,6,2) (b) (1,5,1) (c) (2,1,0)Finding an Equation of a PlaneIn Exercises 3944, find an equation of the plane that passes through the given point and is perpendicular to the given vector or line. PointPerpendicular to (1,3,7) n=j40E41E42EFinding an Equation of a PlaneIn Exercises 3944, find an equation of the plane that passes through the given point and is perpendicular to the given vector or line. PointPerpendicular to (1,4,0) x=1+2t,y=5t,z=32t44EFinding an Equation of a PlaneIn Exercises 4556, find an equation of the plane with the given characteristics. The plane passes through (0,0,0),(2,0,3), and (3,1,5).Finding an Equation of a PlaneIn Exercises 4556, find an equation of the plane with the given characteristics. The plane passes through (3,1,2),(2,1,5), and (1,2,2).Finding an Equation of a PlaneIn Exercises 4556, find an equation of the plane with the given characteristics. The plane passes through (1,2,3),(3,2,1), and (1,2,2).Finding an Equation of a PlaneIn Exercises 4556, find an equation of the plane with the given characteristics. The plane passes through the point (1,2,3) and is parallel to the yz-plane.Finding an Equation of a PlaneIn Exercises 4556, find an equation of the plane with the given characteristics. The plane passes through the point (1,2,3) and is parallel to the xy-plane.50EFinding an Equation of a PlaneIn Exercises 4556, find an equation of the plane with the given characteristics. The plane contains the lines given by x12=y4=zandx23=y14=z21.52EFinding an Equation of a PlaneIn Exercises 4556, find an equation of the plane with the given characteristics. The plane passes through the points (2,2,1) and (1,1,1) and is perpendicular to the plane 2x3y+z=3.54E55E56E57E58EFinding an Equation of a PlaneIn Exercises 5760, find an equation of the plane that contains all the points that are equidistant from the given points. (3,1,2),(6,2,4)60EParallel PlanesIn Exercises 6164, determine whether the planes are parallel or identical. 5x+2y8z=615x6y+24z=1762E63E64EIntersection of PlanesIn Exercises 6568, (a) find the angle between the two planes and (b) find a set of parametric equations for the line of intersection of the planes. 3x+2yz=7x4y+2z=066E67E68EComparing PlanesIn Exercises 6974, determine whether the planes are parallel, orthogonal, or neither. If they are neither parallel nor orthogonal, find the angle between the planes. 5x3y+z=4x+4y+7z=170E71E72E73E74E75E76E77E78E79E80E81E82EIntersection of a Plane and a LineIn Exercises 8386, find the point(s) of intersection (if any) of the plane and the line. Also, determine whether the line lies in the plane. x+3yz=6,x+72=y4=z+1584EIntersection of a Plane and a LineIn Exercises 8386, find the point(s) of intersection (if any) of the plane and the line. Also, determine whether the line lies in the plane. 2x+3y=10,x13=y+12=z386E87E88E89E90E91E92E93E94E95E96E97E98E99E100E101E102E103E104E105E106E107E108E109E110E111E112E113ETrue or False? In Exercises 113118, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. Two lines in space are either intersecting or parallel.115E116E117E118ECONCEPT CHECK Quadric Surfaces How are quadric surfaces and conic sections related?2E3ECONCEPT CHECK Think About It Does every second-degree equation in x, y, and z represent a quadric surface? Explain.5EMatching In Exercises 5-10, match the equation with its graph. [The graphs are labeled (a), (b), (c), (d), (e), and (F).] 15x24y2+15z2=47EMatching In Exercises 5-10, match the equation with its graph. [The graphs are labeled (a), (b), (c), (d), (e), and (F).] y2=4x2+9z29EMatching In Exercises 5-10, match the equation with its graph. [The graphs are labeled (a), (b), (c), (d), (e), and (F).] 4x2y2+4z=0Sketching a Surface in SpaceIn Exercises 1114, describe and sketch the surface. y2+z2=912E13E14E15E16E17E18E19E20E21E22E23E24E25E26E27E28E29E30E31EFinding an Equation for a Surface of RevolutionIn Exercises 3136, find an equation for the surface of revolution formed by revolving the curve in the indicated coordinate plane about the given axis. Equation of Curve Coordinate Plane Axis of Revolution z2=9y yz-plane y-axis33E34E35E36EFinding a Generating CurveIn Exercises 3740, find an equation of a generating curve given the equation of its surface of revolution. x2+y22z=038E39E40E41E42E43EAnalyzing a TraceIn Exercises 43 and 44, analyze the trace when the surface z=12x2+14y2 is intersected by the indicated planes. Find the coordinates of the focus of the parabola formed when the surface is intersected by the planes given by (a) y=4 and (b) x=2.45E46E47E48EUsing a Hyperbolic ParaboloidDetermine the intersection of the hyperbolic paraboloid z=y2b2x2a2 With the plane bx+ayz=0. (Assume a,b0.)50E51ECONCEPT CHECK Cylindrical CoordinatesDescribe the cylindrical coordinate system in your own words.2E3E4E5E6E7E