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mathematics
calculus 10th edition

Calculus 10th Edition Ron Larson, Bruce H. Edwards - Solutions

Three forces with magnitudes of 400 newtons, 280 newtons, and 350 newtons act on an object at angles of -30°, 45°, and 135°, respectively, with the positive x-axis. Find the direction and magnitude of the resultant force.
Prove the Cauchy-Schwarz Inequality, u v ≤ ||u|| ||v||.
Prove the triangle inequality ||u + v|| ≤ ||u|| + ||v||.
In Exercises(a) Find the angle between the two planes(b) Find a set of parametric equations for the line of intersection of the planes. 6x3y + z = 5 -x + y + 5z = 5
Three forces with magnitudes of 75 pounds, 100 pounds, and 125 pounds act on an object at angles of 30°, 45°, and 120°, respectively, with the positive x-axis. Find the direction and magnitude of the resultant force.
In Exercises(a) Find the angle between the two planes(b) Find a set of parametric equations for the line of intersection of the planes. 3x + 2y z = 7 x - 4y + 2z = 0
Forces with magnitudes of 500 pounds and 200 pounds act on a machine part at angles of 30° and -45°, respectively, with the x-axis (see figure). Find the direction and magnitude of the resultant force. 1111 30° -45° 200 lb 500 lb
In Exercises convert the point from spherical coordinates to cylindrical coordinates. 6, 6 이크
In Exercises find the magnitude of v.v = 4i + 3j + 7k
In Exercises determine whether any of the planes are parallel or identical. P₁: 60x + 90y + 30z = 27 P₂: 6x9y3z = 2 P3: 20x + 30y + 10z = 9 P₁: 12x - 18y + 6z = 5
In Exercises find the component form of v given the magnitudes of u and u + v and the angles that u and u + v make with the positive x-axis. u[ = 4, 0 = 30° u + v[ = 6, 0 = 120°
In Exercises convert the point from spherical coordinates to cylindrical coordinates. 18,
In Exercises find the magnitude of v.v = i - 2j - 3k
In Exercises determine whether any of the planes are parallel or identical. P₁: 3x - 2y + 5z = 10 P₂: 6x + 4y - 10z = 5 P3:3x + 2y + 5z = 8 P: 75x50y + 125z = 250 -
In Exercises find the magnitude of v.v = 2i + 5j - k
Consider the vectors u = (cos α, sin α, 0) and v = (cos β, sin β, 0), where a > β. Find the dot product of the vectors and use the result to prove the identitycos(α - β) = cos a cos β + sin α sin β.
In Exercises find the magnitude of v.V = 3j - 5k
In Exercises find the component form of v given the magnitudes of u and u + v and the angles that u and u + v make with the positive x-axis. ||u|| = 1, 0 = 45° ||u + v= √√√2,0 = 90°
In Exercises find the magnitude of v. V v = (1, 0, 3)
In Exercises find a unit vector(a) Parallel to (b) Perpendicular to the graph of ƒ at the given point. Then sketch the graph of ƒ and sketch the vectors at the given point. f(x) = tan x, 7,1
In Exercises convert the point from spherical coordinates to cylindrical coordinates. 36, п,
In Exercises determine whether any of the planes are parallel or identical. P₁: 2xy + 3z = 8 P₂: 3x - 5y2z = 6 P3: 8x 4y + 12z = 5 P₁: PA 4x2y + 6z = 11
In Exercises find the magnitude of v. v = (0, 0, 0)
In Exercises convert the point from spherical coordinates to cylindrical coordinates. 4, ㅠ TT 18' 2.
In Exercises determine whether any of the planes are parallel or identical. P₁: -5x + 2y = 8z = 6 - P₂: 15x6y + 24z = 17 P3: 6x4y + 4z = 9 P₁: 3x - 2y 2z = 4 -
In Exercises convert the point from spherical coordinates to cylindrical coordinates. 10, 6 6' 2
Use vectors to prove that a parallelogram is a rectangle if and only if its diagonals are equal in length.
In Exercises find a unit vector(a) Parallel to (b) Perpendicular to the graph of ƒ at the given point. Then sketch the graph of ƒ and sketch the vectors at the given point. f(x) = 25 x², (3, 4)
Use vectors to prove that the diagonals of a rhombus are perpendicular.
In Exercises(a) Find all points of intersection of the graphs of the two equations(b) Find the unit tangent vectors to each curve at their points of intersection(c) Find the angles (0° ≤ θ ≤ 90°) between the curves at their points of intersection. (y + 1)² = x, y = x³² - 1
In Exercises find a unit vector(a) Parallel to (b) Perpendicular to the graph of ƒ at the given point. Then sketch the graph of ƒ and sketch the vectors at the given point. f(x) = x³, (-2,-8)
In Exercises use vectors to show that the points form the vertices of a parallelogram. (1, 1, 3), (9, -1, -2), (11, 2, -9), (3, 4,-4)
In Exercises convert the point from cylindrical coordinates to spherical coordinates. 4, 3
In Exercises sketch a graph of the plane and label any intercepts.z = 8
Give the standard equation of a sphere of radius r, centered at (x0, Y0, Z0).
Give the standard equation of a plane in space. Describe what is required to find this equation.
A coast artillery gun can fire at any angle of elevation between 0° and 90° in a fixed vertical plane. If air resistance is neglected and the muzzle velocity is constant (= v0), determine the set H of points in the plane andabove the horizontal which can be hit.
Let A, B, and C be vertices of a triangle. Find AB + BC + CA.
State the definition of parallel vectors.
Let r = (x, y, z) and r0 = (1, 1, 1). Describe the set of all points (x, y, z) such that ||r - ro|| = 2.
Describe a method of finding the line of intersection of two planes.
Find the component form of the unit vector v in the direction of the diagonal of the cube shown in the figure. 1 = || A|| V Z X
Describe a method for determining when two planes, a₁x + b₁y + c₁z + d₁ = 0 and a₂x + b₂y + C₂z+d₂ = 0, are (a) Parallel(b) Perpendicular. Explain your reasoning
Let L1 and L2 be nonparallel lines that do not intersect. Is it possible to find a nonzero vector v such that v is normal to both L1 and L2? Explain yourreasoning.
The guy wire supporting a 100-foot tower has a tension of 550 pounds. Using the distances shown in the figure, write the component form of the vector F representing the tension in the wire. X 100 -50 75 N AM y
Match the general equation with its graph. Then state what axis orplane the equation is parallel to.(a)(b)(c)(d)(i)(ii)(iii)(iv) ax + by + d = 0
Personal consumption expenditures (in billions of dollars) for several types of recreation from 2005 through 2010 are shown in the table, where x is the expenditures on amusement parks and campgrounds, y is the expenditures on live entertainment (excluding sports), and z is the expenditures on
The figure shows a chute at the top of a grain elevator of a combine that funnels the grain into a bin. Find the angle between two adjacent sides. 8 in. 6 in. 8 in. 6 in. 8 in.
Find the tension in each of the supporting cables in the figure when the weight of the crate is 500 newtons. 45 cm 65 cm X D A Z C B 70 cm 60 cm 115 cm
Two insects are crawling along different lines in three-space. At time t (in minutes), the first insect is at the point (x, y, z) on the line x = 6 + t, y = 8 t, z = 3 + t. Also, at time t, the second insect is at the point (x, y, z) on the line x = 1 + t, y = 2 + t, z = 2t. Assume that distances
A precast concrete wall is temporarily kept in its vertical position by ropes (see figure). Find the total force exerted on the pin at position A. The tensions in AB and AC are 420 pounds and 650 pounds. Ꭰ 18 ft 6 ft B HOCYKLOWTH AWA C 10 ft → F 8 ft
Find the standard equation of the sphere with center (-3, 2, 4) that is tangent to the plane given by 2x + 4y - 3z = 8.
Write an equation whose graph consists of the set of points P(x, y, z) that are twice as far from A(0, -1, 1) as from B(1, 2, 0). Describe the geometric figure represented by the equation.
Find the point of intersection of the plane 3x - y + 4z = 7 and the line through (5, 4, -3) that is perpendicular to this plane.
Show that the plane 2x - y - 3z = 4 is parallel to the line x = −2 + 2t, y = -1 + 4t, z = 4, and find the distance between them.
Finding a Point of Intersection Find the point of intersection of the line through (1, −3, 1) and (3, -4, 2) and the plane given by x - y + z = 2.
Find a set of parametric equations for the line passing through the point (1, 0, 2) that is parallel to the plane given by x + y + z = 5 and perpendicular to the line x = t, y = 1 + t, z = 1+ t.
In Exercises determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.If v = a1i + b1j + c1k is any vector in the plane given bya₂x + b₂y + c₂z + d₂ = 0, then a₁a₂ + b₁b₂ + c₁c₂ = 0.
In Exercises determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.Every two lines in space are either intersecting or parallel.
In Exercises determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.Two planes in space are either intersecting or parallel.
In Exercises determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.If two lines L1 and L2 are parallel to a plane P, then L1 and L2 are parallel.
In Exercises determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.Two planes perpendicular to a third plane in space are parallel.
In Exercises determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.A plane and a line in space are either intersecting or parallel.
In Exercises find (a) u · v(b) u · u(c) (d) (u · v) v(e) u · (2v) || u ²,
In Exercises convert the point from cylindrical coordinates to rectangular coordinates. (0.5 1.8) 47 3
In Exercises match the equation with its graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).](a)(b)(c)(d)(e)(f) 3 6 4 Z 56 У
In Exercises find (a) u · v(b) u · u(c) (d) (u · v) v(e) u · (2v) || u ²,
In Exercises convert the point from cylindrical coordinates to rectangular coordinates. (3.1)
In Exercises find the cross product of the unit vectors and sketch your result.k × i
In Exercises find the cross product of the unit vectors and sketch your result.j × k
In Exercises match the equation with its graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).](a)(b)(c)(d)(e)(f) 3 6 4 Z 56 У
In Exercises determine whether each point lies on the line.(a) (0, 6, 6)(b) (2, 3, 5) x = -2 + 1, y = 3t, z = 4 + t
In Exercises find (a) u · v(b) u · u(c) (d) (u · v) v(e) u · (2v) || u ²,
In Exercises determine whether each point lies on the line.(a) (7, 23, 0)(b) (1,-1, -3) x-3_y-7 2 8 =2+2
In Exercises match the equation with its graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).](a)(b)(c)(d)(e)(f) 3 6 4 Z 56 У
In Exercises convert the point from cylindrical coordinates to rectangular coordinates.(-7, 0, 5)
In Exercises find the cross product of the unit vectors and sketch your result.j × i
In Exercises find the cross product of the unit vectors and sketch your result.i × j
In Exercises find the coordinates of the point.The point is located three units behind the yz-plane, four units to the right of the xz-plane, and five units above the xy-plane.
In Exercises find the vectors u and v whose initial and terminal points are given. Show that u and v are equivalent. u: (-4, 0), (1,8) v: (2, 1), (7, 7)
In Exercises (a) Find the componentform of the vector v (b) Sketch the vector with its initialpoint at the origin 4 3 نرا 2 1 -1 y H (1, 2) + + 1 2 3 4 (5, 4) 5 X
In Exercises plot the points in the same three-dimensional coordinate system.(a) (2, 1, 3)(b) (-1, 2, 1)
In Exercises plot the points in the same three-dimensional coordinate system.(a) (3, -2, 5)(b) (3/2, 4, -2)
In Exercises (a) Find the component form of the vector v (b) Sketch the vector with its initial point at the origin 4 32 1 -1 -2 y 12 (3,4) V 456 (3,-2)
In Exercises plot the points in the same three-dimensional coordinate system.(a) (5, -2, 2)(b) (5, -2, -2)
In Exercises (a) Find the component form of the vector v (b) Sketch the vector with its initial point at the origin -4 -2 2 (-4,-3) y V −6+ -6 + 2 4 (2, -3)
In Exercises (a) Find the component form of the vector v (b) Sketch the vector with its initial point at the origin (-1,3) T -2 -1 4 2 1 (2, 1) 12 x
In Exercises plot the points in the same three-dimensional coordinate system.(a) (0, 4, -5)(b) (4, 0, 5)
In Exercises find the vectors u and v whose initial and terminal points are given. Show that u and v are equivalent. u: (3, 2), (5, 6) v: (1,4), (3, 8)
In Exercises find the vectors u and v whose initial and terminal points are given. Show that u and v are equivalent. u: (-4,-1), (11,-4) v: (10, 13), (25, 10)
In Exercises find the vectors u and v whose initial and terminal points are given. Show that u and v are equivalent. u: (0, 3), (6, -2) v: (3, 10), (9,5)
In Exercises the initial and terminal points of a vector v are give(a) Sketch the given directed line segment(b) Write the vectorin component form(c) Write the vector as the linearcombination of the standard unit vectors i and j(d) Sketch the vector with its initial point at the origin.
In Exercises the initial and terminal points of a vector v are give(a) Sketch the given directed line segment(b) Write the vector in component form(c) Write the vector as the linear combination of the standard unit vectors i and j(d) Sketch the vector with its initial point at the origin. Initial
In Exercises find the coordinates of the point.The point is located seven units in front of the yz-plane, two units to the left of the xz-plane, and one unit below the xy-plane.
In Exercises the initial and terminal points of a vector v are give(a) Sketch the given directed line segment(b) Write the vector in component form(c) Write the vector as the linear combination of the standard unit vectors i and j(d) Sketch the vector with its initial point at the origin.
In Exercises find the coordinates of the point.The point is located on the x-axis, 12 units in front of the yz-plane.
In Exercises the initial and terminal points of a vector v are give(a) Sketch the given directed line segment(b) Write the vector in component form(c) Write the vector as the linear combination of the standard unit vectors i and j(d) Sketch the vector with its initial point at the origin. Initial
In Exercises find the coordinates of the point.The point is located in the yz-plane, three units to the right of the xz-plane, and two units above the xy-plane.
What is the z-coordinate of any point in the xy-plane?
In Exercises determine the location of a point (x, y, z) that satisfies the condition(s). NIG || N

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