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mathematics
calculus

Questions and Answers of Calculus

The curve defined by x = a cos t, y = a sin t, z = ct is a helix. Hold a fixed and use a CAS to obtain a parmctric plot of the helix for various values of c. What effect does c have on the curve?
For the helix described in Problem 47, hold c fixed and use a CAS to obtain a parametric plot for various values of a. What effect dots a have on the curve?
Find the distance between the following pairs of points. a. (6, - 1, 0) and (1, 2, 3) b. (-2, -2, 0) and (2, -2, -3) c. (e, π, 0) and (-π, - 4, √3)
Show that (4, 5, 3), (1, 7, 4), and (2, 4, 6) are vertices of an equilateral triangle.
Show that (2, 1, 6), (4, 7, 9), and (8, 5, -6) are vertices of a right triangle.
Find the distance from (2, 3, -1) to a. The xy-plane, b. The y-axis, and c. The origin.
A rectangular box has its faces parallel to the coordinate planes and has (2, 3, 4) and (6, -1, 0) as the end points of a main diagonal. Sketch the box and find the coordinates of all eight vertices.
In Problems 1-4 draw the vector w.1. w = u + 3 / 2 v2. w = 2u - 3v 3. w = u1 + u2 + u3 4. w - u1 + u2 + u3
For the two-dimensional vector u and v in Problems a to b, find the sum u + v, the difference u - v, and the magnitudes ||u|| and ||v||. a. u = (-1, 0, 0), v = (3, 4, 0) b. u = (0, 0, 0), v = (-3, 3,
In Figure 18, forces u and v each have magnitude 50 pounds. Find the magnitude and direction of the force w needed to counter balance u and v.
Mark pushes on a post in the direction S 30° E air east of south) with a force of 60 pounds. Dan pushes on the same post in the direction S 60° W with a force of 80 pounds. What are the magnitude
A 300-newton weight rests on a smooth (friction negligible) inclined plane that makes an angle of 30° with the horizontal. What force parallel to the plane will just keep the weight from sliding
An object weighing 258.5 pounds is held in equilibrium by two ropes that make angles of 27.34° and 39.22°, respectively, with the vertical. Find the magnitude of the force exerted on the object by
A wind with velocity 45 miles per hour is blowing in the direction N 20° W. An airplane that flies at 425 miles per hour in still air is supposed to fly straight north. How should the airplane be
A ship is sailing due south at 20 miles per hour. A man walks west (i.e., at right angles to the side of the ship) across the deck at 3 miles per hour. What are the magnitude and direction of his
Julie, flying in a wind blowing 40 miles per hour due south, discovers that she is heading due east when she points her airplane in the direction N 60c E. Find the airspeed (speed in still air) of
What heading and airspeed are required for an airplane to fly 837 miles per hour due north if a wind of 63 miles per hour is blowing in the direction S 11.5° E?
Prove, using vector methods, that the line segment joining the midpoints of two sides of a triangle is parallel to the third side.
Prove that the midpoints of the four sides of an arbitrary quadrilateral are the vertices of a parallelogram.
Let v1, v2, . . . .,vn be the edges of a polygon arranged in cyclic order as shown for the cas n = 7 in Figure 19. Show thatv1, v2 + .... + vn = 0
Let n points be equally spaced on a circle, and let v1, v2,..., vn be the vectors from the center of the circle to these n points. Show that v1, v2 +. . . . + vn = 0.
Consider a horizontal triangular table with each vertex angle less than 120°. At the vertices are frictionless pulleys over which pass strings knotted at P, each with a weight W attached as shown
Show that the point P of the triangle of Problem 31 that minimizes |AP| + |BP| + |CP| is the point where the three angles at P are equal. Let A', B', and C' be the points where the weights are
Let the weights at A, B, and C of Problem 31 be 3w, 4w, and 5w, respectively. Determine the three angles at P at equilibrium. What geometric quantity (as in Problem 32) is now minimized?
A 100-pound chandelier is held in place by four wires attached to the ceiling at the four corners of a square. Each wire makes an angle of 45° with the horizontal. Find the magnitude of the tension
Repeat Problem 35 for the case where there are three wires attached to the ceiling at the three corners of an equilateral triangle. A 100-pound chandelier is held in place by four wires attached to
For the two-dimensional vector u and v in Problems a-b, find the sum u + v, the difference u - v, and the magnitudes ||u|| and ||v||. a. u = (-1, 0), v = (3, 4) b. u = (0, 0), v = (-3, 4)
Let a = -2i + 3j, b = 2i - 3j, and = -5j. Find each of the following: a. 2a - 4b b. a ∙ b c. a ∙ (b + c) d. (-2a + 3b) ∙ 5c e. ||a||c ∙ a f. b ∙ b - ||b||
For the vectors a, b, and c from Problem 8, find the direction cosines and the direction angles. In Problem 8 Let a = (3√/3, √3/3, √3/3), b = (1, -1 0), and c = (-2, -2, 1). Find the angle
Show that the vectors a = (1, 1, 1), b = (1, -1, 0), and c = (-1, -1, 2) are mutually orthogonal, that is, each pair of vectors is orthogonal.
Show that the vectors a = i - j, b = i + j, and c = 2k are mutually orthogonal, that is each pair of vectors is orthogonal.
If u + v is orthogonal to u - v, what can you say about the relative magnitudes of u and v?
Find two vectors of length 10, each of which is perpendicular to both -4i + 5j + and 4i + j.
Find all vectors perpendicular to both (1, -2, -3) and (-3, 2, 0).
Find the angle ABC if the points are A(1, 2, 3), B( -4, 5, 6), and C(1, 0, 1).
Show that the triangle ABC is a right triangle if the vertices are A(6, 3,3), B(3, 1, -1), and C(-1, 10, -2.5).
Let a = (3, -1), b = (1, -1), and c = (0, 5). Find each of the following: a -4a + 3b b. b ∙ c c. (a + b) ∙ c d. 2c ∙ (3a + 4b) e. ||b||b ∙ a f. ||c||2 - c ∙ c
For what values of a, b, and c are the three vectors (a, 0, 1), (0, 2, b), and (1, c, 1) mutually orthogonal.
In Problems a-c, find each of the given projections if u = i + 2j, v = 2i - j, and w = i + 5j. a. projv u b. proju v c. proju w
In Problems a to c, find each of the given projections if u = 3i + 2j + k, v = 2i - k, and w = i + 5j - 3k. a. projv u b. profu v c. profu w
Find the cosine of the angle between a and b and make a sketch. a. a = (1, -3), b = (-1, 2) b. a = (-1, -2), b = (6, 0) c. a = (2, -1), b = (-2, -4) d. a = (4, -7), b = (-8, 10)
Find a simple expression for each of the following for an arbitrary vector u. a. proju u b. proj-u u
Find a simple expression for each of the following for an arbitrary vector u. a. proju (-u) b. proj-u (-u)
Find the scalar projection of u = + 5j + 3k on v = + j - k.
Find the scalar projection of is u = 5i + 5j + 2k on v = - √5i + √5j + k.
Find the angle between a and b and make a sketch. a. a = 12i, b = -5i b. a = 4i + 3j. b = -8i - 6j c. a = + 3j, b = 2i - 6j d. a = √3j + j, b = 3i + √3j
If α = 46° and β = 108° are direction angles for a vector u, find two possible values for the third angle.
Find two perpendicular vectors u and v such that each is also perpendicular to w = (-4, 2, 5).
Find the vector emanating from the origin whose terminal point is the midpoint of the segment joining (3, 2, -1) and (5, -7, 2).
In Problems a-c, give a proof of the indicated property for two-dimensional vectors. Use u = (u1, u2), v = (v1, v2), and w = (w1, w2). a. (a + b)u = au + bu b. u ∙ v = v ∙ u c. c(u ∙ v) = (cu)
Le a = i + 2j - k, b = j + k, and c = -i + j + 2k. Find the each of following: a. a ∙ b b. (a + c) ∙ b c. a / ||a|| d. (b - c) ∙ a e. a ∙ b / ||a|| ||b|| f. b ∙ b - ||b||2
Given the two nonparallel vectors a = 3i - 2j and b = -3i 4j and another vector r = 7i - 8j, find scalars k and m such that r = ka + mb.
Given the two nonparallel vectors a = -4i + 3j and b = 2i - j and another vector r = 6i - 7j, find scalars k and m such that r = ka + mb.
Show that the vector n = al t bj is perpendicular to the line with equation ax + by = c. Let P1(x1, y1) and P2(x2, y2) be two points on the line and show that n ∙ = 0.
Prove that ||u + v||2 + ||u - v||2 = 2||u||2 + 2||v||2.
Prove that u ∙ v = 1/4 ||u + v||2 v||2 - 1/4 ||u - v||2.
Find the angle between a main diagonal of a cube and one of its faces.
Find the smallest angle between the main diagonals of a rectangular box 4 feet by 6 feet by 10 feet.
Find the angles formed by the diagonals of a cube.
Let a = (√2, √2, 0), b = (1, -1, 1), and c = (-2, 2, 1). Find each of the following: a. a ∙ c b. (a - c) ∙ b c. a / ||a|| d. (b - c) ∙ a e. b ∙ c / ||b|| ||c|| f. a ∙ a - ||a||2
Find the work done by a force of 100 newtons acting in the direction S 70° E in moving an object 30 meters east.
Find the work done by a force F = -4k newtons in moving an object from (0. 0, 8) to (4, 4, 0), where distance is in meters.
Find the work done by a force F = 3i - 6j + 7k pounds in moving an object from (2, 1, 3) to (9, 4, 6), where distance is in feet.
In Problems a-d, find the equation of the plane having the given normal vector n and passing through the given point P. a. n = 2i - 4j + 3k; F(1, 2, -3) b. n = 3i - 2j - l k; P(-2, -3, 4) c. n = (1,
For the vectors a, b, and c from Problem 6, find the angle between each pair of vectors. In Problem 6 Let a = (√2, √2, 0), b = (1, -1, 1), and c = (-2, 2, 1).
Find the equation of the plane through (-1, 2, -3) and parallel to the plane 2x + 4y - z = 6.
Find the equation of the plane passing through (-4, -1, 2) and parallel to a. The xy-plane b. The plane 2x - 3y - 4z = 0
Find the equation of the plane passing through the origin and parallel to a. The xy-plane b. The plane x + y + z = 1
Find the distance between the parallel planes + 2y + z = 9 and 6x - 4y - 2z = 19.
Find the distance between the parallel planes 5x - 3y - 2z = 5 and -5x + 3y + 2z = 7.
Find the distance from the sphere x2 + y2 + z2 + 2x + 6y - 8z = 0 to the plane 3x + 4y z = 15.
Find the equation of the plane each of whose points is equidistant from (-2, 1. 4) and (6, 1, -2).
Let a = (3√/3, √3/3, √3/3), b = (1, -1 0), and c = (-2, -2, 1). Find the angle between each pair of vectors
Prove the Mangle Inequality see Figure 13 for two-dimensional vectors:||u + v|| ‰¤ ||u|| + ||v||:
A weight of 30 pounds is suspended by three wires with resulting tensions 3i + 4j + 15k, - 8i - 2j + 10k, and ai + bj + ck. Determine a, b, and c so that the net force is straight up.
Show that the work done by a constant force F on an object that moves completely around a closed polygonal path is 0.
Let a = (a1, a2, a3) and b = (b1, b2, b3) be fixed vectors. Show that (x - a) ∙ (x - b) = 0 is the equation of a sphere, and find its center and radius.
Refine the method of example 10 by showing that distance L between the parallel planes Ax + By + Cz = D and Ax + By + Cz = E is
The medians of a triangle meet at a point P (the centroid by Problem 30 of Section 5.6) that is two-thirds of the way from a vertex to the midpoint of the opposite edge. Show that p is the head of
Suppose that the three coordinate planes bounding the first octant are mirrors. A light ray with direction ai + bj + ck is reflected successively from the xy-plane, the xz-plane, and the yz-plane.
For the vectors a, b, and c from Problem 6, find the direction cosines and the direction angles. In Problem 6 Let a = (√2, √2, 0), b = (1, -1, 1), and c = (-2, 2, 1).
Let a = -3i + 2k - 2l, b = -i + 2j - 4k, and c = 7i + 3j - 4k. Find each of the following: a. a × b b. a × (b + c) c. a ∙ (b + c) d. a × (b × c)
Find the area of the triangle with (1, 2, 3), (3, 1, 5), and (4, 5, 6) as vertices.
In Problems a-b, find the equation of the plane through the given points. a. (1, 3, 2), (0, 3, 0), and (2, 4, 3) b. (1, 1, 2), (0, 0, 1), and (-2, -3, 0)
Find the equation of the plane through (2, 5, 1) that is parallel to the plane x - y + 2z = 4.
Find the equation of the plane through (0, 0, 2) that is parallel to the plane x + y + z = 1.
Find the equation of the plane through (-1, -2, 3) and perpendicular to both the planes x - 3y + 2z = 7 and 2x - 2y - z = -3.
Find the equation of the plane through (2, -1, 4) that is perpendicular to both the planes x + y + z = 2 and x - y - z = 4.
Find the equation of the plane through (2, -3, 2) and parallel to the plane of the vectors 4i + 3j - k and 2i - 5j + 6k.
If a = (3, 3, 1), b = (-2, -1, 0), and c = (-2, -3, -1), find each of the following: a. a × b b. a × (b +c) c. a ∙ (b × c) d. a × (b × c)
Find the equation of the plane through the origin that is perpendicular to the xy-plane and the plane 3x - 2y + z = 4.
Find the equation of the plane through (6, 2, -1) and perpendicular to the line of the planes 4x - 3y + 2z + 5 = 0 and 3x + 2y - z + 11 = 0.
Let a and b be nonparallel vectors, and let c be any nonzero vector. Show that (a × b) × e is a vector in the plane of a and b.
Let K be the parallelepiped determined by u = (3, 2, 1), v = (1, 1, 2), and w = (1, 3, 3). a. Find the volume of K. b. Find the area of the face determined by u and v. c. Find the angle between u and
The formula for the volume of a parallelepiped derived in Example 4 should not depend on the choice of which one of the tree vectors we call a, which one we call b, and which one we call c. Use this
The volume f a tetrahedron is known to be 1/3(area of base) (height). From this, show that the volume of the tetrahedron with edges a, b, and c is 1/6 |a ∙ (b × c)|.
Find all vectors perpendicular to both of the vectors a = i + 2j + 3k and b = -2i + 2j - 4k.
Find the volume of the tetrahedron with vertices (-1, 2, 3), (4, -1, 2), (5, 6, 3), and (1, 1, -2)
Prove Lagrange's Identity, ||u × v||2 = ||u||2||v||2 - (u × v)2
Prove the left distributive law. u × (v + w) = (u × v) + (u × w)

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