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mathematics
calculus 10th edition
Calculus 10th Edition Ron Larson, Bruce H. Edwards - Solutions
Find the following.(a)(b)(c)(d)(e)(f) || n ||
Use the shell method to find the volume of the solid below the surface of revolution and above the xy-plane.The curve z = sin y (0 ≤ y ≤ π) in the yz-plane is revolvedabout the z-axis.
Use the triple scalar product to find the volume of the parallelepiped having adjacent edges u = 2i + j, v = 2j + k, and w = -j + 2k.
Find an equation of the plane passing through the point perpendicular to the given vector or line. Point (-1,4, 0) Perpendicular to x = -1 + 2t, y = 5-t, z = 3 - 2t
Convert the point from spherical coordinates to rectangular coordinates. 5, 4' З п 4
In Exercises (a) Find the projection of u onto v(b) Find the vector component of u orthogonal to v u = (8, 2, 0), v = (2, 1, -1)
Find the standard equation of the sphere. Center: (-3, 2, 4), tangent to the yz-plane
Three types of classic “topological” surfaces are shown below. The sphere and torus have both an “inside” and an “outside.” Does the Klein bottle have both an inside and an outside? Explain. Sphere Klein bottle Torus Klein bottle
Prove the property of the cross product.u × u = 0
The vector v and its initial point are given. Find the terminal point. v = (1, -3,1) Initial point: (0, 2, 1)
Prove the property of the cross product.u . (v × w) = (u × v) . w
Describe and sketch the surface.x + 2y + 3z = 6
Find the component form of u + v given the lengths of u and v and the angles that u and v make with the positive x-axis. |u|| = 5, 0₁= -0.5 u ||v|| = 5, 0, 0.5
Match the equation (written in terms of cylindrical or spherical coordinates) with its graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).](a)(b)(c)(d)(e)(f)r = 5 -3-2 X Z 3+ 2 3 元 4 23 y
Describe and sketch the surface.y = z2
Prove the property of the cross product.u × v = 0 if and only if u and v are scalar multiples of eachother.
Determine whether the planes are parallel, orthogonal, or neither. If they are neither parallel nor orthogonal, find the angle of intersection. 5x 3y + z = 4 x + 4y + 7z = 1
Describe and sketch the surface. y = ¹/2 24
Match the equation (written in terms of cylindrical or spherical coordinates) with its graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).](a)(b)(c)(d)(e)(f) -3-2 X Z 3+ 2 3 元 4 23 y
Determine whether the planes are parallel, orthogonal, or neither. If they are neither parallel nor orthogonal, find the angle of intersection. 3x + y - 4z = 3 - - 9x - 3у + 12z = 4
Find sets of (a) Parametric equations(b) Symmetric equations of the line through the two points (if possible). (For each line, write the direction numbers as integers.) (5, -3, -2), (-3, 2, 1)
Describe and sketch the surface.y² + z² = 9
Find an equation in rectangular coordinates for the equation given in cylindrical coordinates, and sketch its graph.z = r² cos² θ
Find a unit vector that is orthogonal to both u = (2, 10, 8) and v = (4, 6, -8).
Find sets of (a) Parametric equations (b) Symmetric equations of the line through the two points.(For each line, write the direction numbers as integers.) (-1, 4, 3), (8, 10, 5)
Find an equation in spherical coordinates for the equation given in rectangular coordinates.z = 6
When u x v = 0 and u • v = 0, what can you conclude about u and v?
Find a set of parametric equations of the line.The line passes through the point (1, 2, 3) and is perpendicular to the xz-plane.
Sketch a graph of u, v, and u + v. Then demonstrate the triangle inequality using the vectors u and v. u = (2, 1), v = (5, 4)
Find an equation of the surface satisfying the conditions, and identify the surface. The set of all points equidistant from the point (0, 2, 0) and the plane y = -2
Complete the square to write the equation of the sphere in standard form. Find the center and radius.9x² + 9y² + 9z² - 6x + 18y + 1 = 0
Find an equation of the plane.The plane passes through (1, 2, 3), (3, 2, 1), and (-1, -2, 2).
When the magnitudes of two vectors are doubled, how will the magnitude of the cross product of the vectors change? Explain.
Sketch a graph of u, v, and u + v. Then demonstrate the triangle inequality using the vectors u and v. u = (-3, 2), v = (1, -2)
The vertices of a triangle in space are (x₁, y₁, Z₁), (x₂, Y2, Z₂), and (X3, y3, Z3). Explain how to find a vector perpendicular to the triangle. 4 X -2 3 2 5 4 3 2 է: (X3, )3, Z3) (Xլ, 1լ, Zլ) 5 (X, Y, Z2) y
Find an equation in spherical coordinates for the equation given in rectangular coordinates.x2 + y2 + z2 = 49
Determine which of the following are defined for nonzero vectors u, v, and w. Explain your reasoning.(a) u . (v + w)(b) (u . v)w(c) u. v + w (d) ||u|| (v + w)
State the definition of orthogonal vectors. When vectors are neither parallel nor orthogonal, how do you find the angle between them? Explain.
Find an equation of the plane. The plane passes through the point (1, 2, 3) and is parallel to the yz-plane.
Complete the square to write the equation of the sphere in standard form. Find the center and radius.4x² + 4y² + 4z² - 24x - 4y + 8z - 23 = 0
In Exercises(a) Find the component form of the vector v(b) Write the vector using standard unit vector notation(c) Sketch the vector with its initial point at the origin (4, 2, 1) 67 X 6 4 2 Z V , (2, 4, 3) 6 y
Find an equation of the surface satisfying the conditions, and identify the surface. The set of all points equidistant from the point (0, 0, 4) and the xy-plane
Find a set of parametric equations of the line. The line passes through the point (1, 2, 3) and is parallel to the line given by x = y = z.
Find the vector with the given magnitude and the same direction as u. Magnitude ||v|| = 6 Direction u = (0, 3)
State the geometric properties of the cross product.
Because of the forces caused by its rotation, Earth is an oblate ellipsoid rather than a sphere. The equatorial radius is 3963 miles and the polar radius is 3950 miles. Find an equation of the ellipsoid. (Assume that the center of Earth is at the origin and that the trace formed by the plane z = 0
Find an equation in spherical coordinates for the equation given in rectangular coordinates.x² + y² - 3z² = 0
Find an equation of the plane. The plane passes through the point (1, 2, 3) and is parallel to the xy-plane.
Find a set of parametric equations of the line. The line is the intersection of the planes 3x - 3y - 7z = − 4 and x - y + 2z = 3.
Find the vector with the given magnitude and the same direction as u. Magnitude ||v|| = 4 Direction u = (1, 1)
Find an equation in spherical coordinates for the equation given in rectangular coordinates.x² + y² = 16
The top of a rubber bushing designed to absorb vibrations in an automobile is the surface of revolution generated by revolving the curvefor 0 ≤ y ≤ 2 in the yz-plane about the z-axis.(a) Find an equation for the surface of revolution.(b) All measurements are in centimeters and the bushing is
Describe direction cosines and direction angles of a vector v.
Find an equation of the plane. x - 1 -2 and =y=4 = z *-32-21-22 y 4 -
Find an equation of the plane.The plane contains the y-axis and makes an angle of π/6 withthe positive x-axis.
Determine the intersection of the hyperbolic paraboloidwith the plane bx + ay - z = 0. (Assume a, b > 0.) ม || Z
In Exercises(a) Find the component form of the vector v(b) Write the vector using standard unit vector notation(c) Sketch the vector with its initial point at the origin 6 X 4 2 6 4 (0,3,3) 21 V 4 (3,3,0) 6
Find a set of parametric equations of the line.The line passes through the point (0, 1, 4) and is perpendicular to u = (2, -5, 1) and v = (-3, 1, 4).
Find the vector with the given magnitude and the same direction as u. Magnitude ||v|| = 5 Direction u = (1, 2)
Find an equation in spherical coordinates for the equation given in rectangular coordinates.x = 13
In Exercises(a) Find the component form of the vector v(b) Write the vector using standard unit vector notation(c) Sketch the vector with its initial point at the origin X 2 61 4 (2, 3, 4) 21 (2,3,0) 4 +y 6
Give a geometric description of the projection of u onto v.
Find an equation of the plane. The plane passes through the point (2, 2, 1) and contains the line given by X 2 y-4 -1 Z.
Find an equation of the plane.The plane passes through (-3, -4, 2), (-3, 4, 1), and (1, 1, -2).
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If u # 0 and u x v = u xw, then v = w.
Find the vector with the given magnitude and the same direction as u. Magnitude ||v|| = 2 Direction 1=(√3,3)
Explain why the curve of intersection of the surfaceslies in a plane. x² + 3y² 2z² + 2y = 4 and 2x² + 6y²4z² - 3x = 2
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.It is possible to find the cross product of two vectors in atwo-dimensional coordinate system.
Find an equation in spherical coordinates for the equation given in rectangular coordinates.x2 + y2 = 2z2
What can be said about the vectors u and when (a) The projection of u onto v equals u (b) Theprojection of u onto v equals 0?
Find the component form and magnitude of the vector v with the given initial and terminal points. Then find a unit vector in the direction of v. Initial point: (3, 2, 0) Terminal point: (4, 1, 6)
Find the component form of given its magnitude and the angle it makes with the positive x-axis. ||v|| = 3, 0 = 0°
When the projection of u onto v has the same magnitude as the projection of v onto u, can you conclude thatExplain. ||u|| = || v ||?
Find an equation of the plane. The plane passes through the point (-2, 3, 1) and is perpendicular to n = 3i - j + k.
Find an equation in spherical coordinates for the equation given in rectangular coordinates.x² + y² + z² - 9z = 0
What is known about θ, the angle between two nonzero vectors u and v,when(a) u . v = 0? (b) u . v > 0? (c) u . v < 0? u 0 Origin
Find an equation of the plane. The plane contains the lines given by x - 1 -2 || yz + 1 and x + 1 -2 = y1=z - 2.
Find an equation in rectangular coordinates for the equation given in spherical coordinates, and sketch its graph.ρ = 5
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If u # 0 and u x v = uxw, then v = w.
Find an equation in rectangular coordinates for the equation given in spherical coordinates, and sketch its graph. 0 = З п 4
Find an equation of the plane. The plane passes through the points (2, 2, 1) and (-1, 1, -1) and is perpendicular to the plane 2x - 3y + z = 3.
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.A sphere is an ellipsoid.
Find the component form and magnitude of the vector v with the given initial and terminal points. Then find a unit vector in the direction of v. Initial point: (1, -2, 4) Terminal point: (2, 4, -2)
Find the component form of given its magnitude and the angle it makes with the positive x-axis. |v|| 5, 0= 120⁰° =
Find an equation of the plane. The plane passes through the points (3, 2, 1) and (3, 1, -5) and is perpendicular to the plane 6x + 7y + 2z = 10.
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If u 0, u vu w, and u x y = u x w, then v = w. . .
Find an equation in rectangular coordinates for the equation given in spherical coordinates, and sketch its graph. El || 9 Φ
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.The generating curve for a surface of revolution is unique.
Prove the property of the cross product. (MXN) + (A x n) = (M + A) x n
The initial and terminal points of a vector v are given.(a) Sketch the directed line segment(b) Find the componentform of the vector(c) Write the vector using standard unitvector notation(d) Sketch the vector with its initial pointat the origin. Initial point: (-1, 2, 3) Terminal point: (3, 3, 4)
Find an equation of the plane.The plane passes through the points (5, 1, 3) and (2, -2, 1) and is perpendicular to the plane 2x + y -z = 4.
Find the component form of given its magnitude and the angle it makes with the positive x-axis. |v|| = 2, 0 = 150⁰
Find an equation of the plane. The plane passes through the points (1, -2, -1) and (2, 5, 6) and is parallel to the x-axis.
The vector u = (3240, 1450, 2235) gives the numbers of hamburgers, chicken sandwiches, and cheeseburgers, respectively, sold at a fast-food restaurant in one week. The vector v = (2.25, 2.95, 2.65) gives the prices (in dollars) per unit for the three food items. Find the dot product u . v, and
Find the component form of given its magnitude and the angle it makes with the positive x-axis. |v||4, 03.5°
Find an equation in rectangular coordinates for the equation given in spherical coordinates, and sketch its graph. || Φ
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.All traces of an ellipsoid are ellipses.
The initial and terminal points of a vector v are given.(a) Sketch the directed line segment(b) Find the component form of the vector(c) Write the vector using standard unit vector notation(d) Sketch the vector with its initial point at the origin. Initial point: (2, -1, -2) Terminal point: (-4, 3,
Find the distance between the point (1, 0, 2) and the plane 2x - 3y + 6z = 6.
Find two vectors in opposite directions that are orthogonal to the vector u. (The answers are not unique.) = - + j u =
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