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mathematics
calculus 4th
Calculus 4th Edition Jon Rogawski, Colin Adams, Robert Franzosa - Solutions
Determine whether the two vectors are orthogonal and, if not, whether the angle between them is acute or obtuse. (0, 2,4), (3, 1, 0)
Calculate.5 · 6, 2
Determine whether is equivalent to AB
Determine whether the two vectors are orthogonal and, if not, whether the angle between them is acute or obtuse. 〈一〉〈第一会〉
Calculate.4((1, 1) + (3, 2))
Determine whether is equivalent to AB
Determine whether the two vectors are orthogonal and, if not, whether the angle between them is acute or obtuse. (12,6), (2,-4)
Find the cosine of the angle between the vectors. (0, 3, 1), (4, 0, 0)
Find the cosine of the angle between the vectors. (1, 1, 1), (2,-1,2)
Find the cosine of the angle between the vectors.i + j, j + 2k
Find the angle between the vectors. (2, √2), (1 + √2,1 – √2) -
Find the cosine of the angle between the vectors.3i + k, i + j + k
Find the angle between the vectors. (5, √3), (√3,2)
Find the angle between the vectors. (1, 1, 1), (1,0,1)
Find the angle between the vectors. (3, 1, 1), (2,-4, 2)
Find the angle between the vectors. (0, 1, 1), (1,-1,0)
Find the angle between the vectors. (1, 1,-1), (1, -2, -1)
Find the angle between the vectors.i, 3i + 2j + k
Find the angle between the vectors.i + k, j − k
Find all values of b for which the vectors are orthogonal. (a) (b, 3, 2), (1,b, 1) (b) (4, -2,7), (b², b.0)
Assume that v lies in the yz-plane. Which of the following dot products is equal to zero for all choices of v? (a) v. (0, 2, 1) (c) v. (-3,0,0) V (b) v. k (d) v. j V
Find a vector that is orthogonal to (−1, 2, 2).
Find two vectors that are not multiples of each other and are both orthogonal to (2, 0, −3).
Find a vector that is orthogonal to v = (1, 2, 1) but not to w = (1, 0, −1).
Find v · e, where ΙΙvΙΙ = 3, e, is a unit vector, and the angle between e and v is 2π/3.
Use the properties of the dot product to evaluate the expression, assuming that u · v = 2, ΙΙuΙΙ = 1, and ΙΙvΙΙ = 3.u · (4v)
Use the properties of the dot product to evaluate the expression, assuming that u · v = 2, ΙΙuΙΙ = 1, and ΙΙvΙΙ = 3.(u + v) · v
Use the properties of the dot product to evaluate the expression, assuming that u · v = 2, ΙΙuΙΙ = 1, and ΙΙvΙΙ = 3.2u · (3u − v)
Use the properties of the dot product to evaluate the expression, assuming that u · v = 2, ΙΙuΙΙ = 1, and ΙΙvΙΙ = 3.(u + v) · (u − v)
Find the angle between v and w if v · w = − ||v||||wΙ||.
Find the angle between v and w if v · w = 1/2 ||v|| ||w||.
Assume that v = 2, w = 3, and the angle between v and w is 120◦. Determine: (a) v. w (b) ||2v + w|| (c) ||2v - 3w||
Assume that ΙΙvΙΙ = 3, ΙΙwΙΙ = 5, and the angle between v and w is θ = π/3.(a) Use the relation ΙΙv + w ΙΙ2 = (v + w) · (v + w) to show that ΙΙv + wΙΙ 2 = 32 + 52 + 2v · w.(b) Find ΙΙv + wΙΙ.
Show that if e and f are unit vectors such that |le + f|| = 2, then ||e f|| = 7. Show that e. f = -100
Find the angle θ in the triangle in Figure 17. (0, 10) (3, 2) Ө (10, 8)
Find ||2e − 3f||, assuming that e and f are unit vectors such that ||e + f|| = √3/2.
Find all three angles in the triangle in Figure 18. (0,0) (2,7) (6,3)
(a) Draw uΙΙv and vΙΙu for the vectors appearing as in Figure 19.(b) Which of uΙΙv and vΙΙu has the greater magnitude? V u
Find the projection of u along v. u= (2,5), v= (1, 1)
Let u and v be two nonzero vectors.(a) Is it possible for the component of u along v to have the opposite sign from the component of v along u? Why or why not?(b) What must be true of the vectors if either of these two components is 0?
Find the projection of u along v. u= (2,-3), v= (1,2)
Find the projection of u along v. u= (-1,2,0), v = (2,0, 1)
Find the projection of u along v. u = (1, 1, 1), v = (1,1,0)
Find the projection of u along v.u = 5i + 7j − 4k, v = k
Find the projection of u along v. u= (a, b, c), v=i
Find the projection of u along v.u = i + 29k, v = j
Find the projection of u along v. u = (a, a,b), v=i-j
Compute the component of u along v. u (3, 2, 1), v= (1, 0, 1) =
Compute the component of u along v. u = (3,0,9), v = (1, 2,2)
Find ΙΙu⊥vΙΙ in Figure 20. Ujv Lv u = (3,5) P v = (8, 2)
Find the decomposition a = aΙΙb + a⊥b with respect to b. a = (1,0), b = (1, 1)
Find the decomposition a = aΙΙb + a⊥b with respect to b. a = (2, -3), b = (5,0)
Find the decomposition a = aΙΙb + a⊥b with respect to b. a =(4,-1,0), b = (0, 1, 1)
Find the decomposition a = aΙΙb + a⊥b with respect to b. a (4,-1,5), b = (2, 1,1) =
Find the decomposition a = aΙΙb + a⊥b with respect to b. a = (x, y), b = (1,-1)
Find the decomposition a = aΙΙb + a⊥b with respect to b. a = (x, y, z), b = (1, 1, 1)
Let v and w be vectors in the plane.(a) Use Theorem 2 to explain why the dot product v · w does not change if both v and w are rotated by the same angle θ. THEOREM 2 Dot Product and the Angle Let 9 be the angle between two nonzero vectors v and w. Then vw |v||||w|| cos 6 or cos V. W ||v||||w||
Let eθ = (cos θ, sin θ). Show that eθ · eψ = cos(θ − ψ) for any two angles θ and ψ.
Refer to Figure 21.Find the angle between AB and AC.
Refer to Figure 21.Find the angle between AB and AD.
As in Example 9, assume that the carpenter’s diagonal measurements are x and y, and compute the diagonal length that produces a rectangular frame. Compare the result with the corresponding split-difference approximation.x = 87.2 cm and y = 82.7 cm EXAMPLE 9 Let us return to the problem posed at
Refer to Figure 21.Calculate the projection of AĎ along AB.
Refer to Figure 21.Calculate the projection of AC along AD.
As in Example 9, assume that the carpenter’s diagonal measurements are x and y, and compute the diagonal length that produces a rectangular frame. Compare the result with the corresponding split-difference approximation. 234 inches and y = 223 in.
The methane molecule CH4 consists of a carbon molecule bonded to four hydrogen molecules that are spaced as far apart from each other as possible. The hydrogen atoms then sit at the vertices of a tetrahedron, with the carbon atom at its center, as in Figure 22. We can model this with the carbon
Iron forms a crystal lattice where each central atom appears at the center of a cube, the corners of which correspond to additional iron atoms, as in Figure 23. Use the dot product to find the angle β between the line segments from the central atom to two adjacent outer atoms. Take the central
In Example 9, assume that the wind is out of the north at 45 km/h. Express the corresponding wind vector as a sum of vectors, one parallel to the bridge and one perpendicular to it. Also, compute the magnitude of the perpendicular term to determine the speed of the part of the wind blowing directly
Let v and w be nonzero vectors and set u = ev + ew. Use the dot product to show that the angle between u and v is equal to the angle between u and w. Explain this result geometrically with a diagram.
Let v, w, and a be nonzero vectors such that v · a = w · a. Is it true that v = w? Either prove this or give a counterexample.
Calculate the force (in newtons) required to push a 40-kg wagon up a 10◦ incline (Figure 24). 10⁰ 40 kg
A plane flies with velocity v = (220, −90, 10) km/h. A wind is blowing out of the northeast with velocity w = (−30, −30, 0) km/h. Express the wind vector as a sum of vectors, one parallel to the plane’s velocity, one perpendicular to it. Is the parallel part of the wind blowing with or
A force F is applied to each of two ropes (of negligible weight) attached to opposite ends of a 40-kg wagon and making an angle of 35◦ with the horizontal (Figure 25). What is the maximum magnitude of F (in newtons) that can be applied without lifting the wagon off the ground? F 35⁰ 40
A light beam travels along the ray determined by a unit vector L, strikes a flat surface at point P, and is reflected along the ray determined by a unit vector R, where θ1 = θ2 (Figure 26). Show that if N is the unit vector orthogonal to the surface, then R = 2(L.N)N - L Incoming light L N 01
Let P and Q be antipodal (opposite) points on a sphere of radius r centered at the origin and let R be a third point on the sphere (Figure 27). Prove thatare orthogonal. PR and QR
Prove that ΙΙv + wΙΙ2 − ΙΙv − wΙΙ2 = 4v · w.
A rhombus is a parallelogram in which all four sides have equal length. Show that the diagonals of a parallelogram are perpendicular if and only if the parallelogram is a rhombus. Take an approach similar to the solution in Example 5, and consider x · y. EXAMPLE 5 Figure 7 shows a parallelogram
Use Exercise 91 to show that v and w are orthogonal if and only if ΙΙv − wΙΙ = ΙΙv + wΙΙ.Data From Exercise 91Prove that ΙΙv + wΙΙ2 − ΙΙv − wΙΙ2 = 4v · w.
Verify the Distributive Law: M. n + A. n = (M + A) .n
Prove the Law of Cosines, c2 = a2 + b2 − 2ab cos θ, by referring to Figure 28. Hint: Consider the right triangle ΔPQR. S 0 a a sin 0 b R 1 P C b-a cos 0 Q
Verify that (λv) · w = λ(v · w) for any scalar λ.
Use (7) to prove the Triangle Inequality:First use the Triangle Inequality for numbers to prove ||M|| + ||A|| > ||M + A||
In this exercise, we prove the Cauchy–Schwarz inequality: If v and w are any two vectors, then |vw|≤ ||v||||w|| . |xv +w||2 for x a scalar. Show that f(x) = ax² + bx+c, where a = (a) Let f(x) = c = ||w||². (b) Conclude that b² - 4ac ≤ 0. Observe that f(x) ≥ 0 for all x. ||v||², b 7 =
Let v = (x, y) andProve that the angle between v and vθ is θ. Ve= (x cos 0 + y sin 0, -x sin 0 + y cos 0)
Let v be a nonzero vector. The angles α, β, γ between v and the unit vectors i, j, k are called the direction angles of v (Figure 30). The cosines of these angles are called the direction cosines of v. Prove that cos² a + cos² B+ cos² y = 1
The set of all points X = (x, y, z) equidistant from two points P, Q in R3 is a plane (Figure 31). Show that X lies on this plane if PQ-OX = -(100|1²- ||OPIP)
Find the direction cosines of v = (3, 6, −2).
Use Eq. (8) to find the equation of the plane consisting of all points X = (x, y, z) equidistant from P = (2, 1, 1) and Q = (1, 0, 2). 1 PÒ · OX = — (11081||² – ||0²|1²³)
Sketch the plane consisting of all points X = (x, y, z) equidistant from the points P = (0, 1, 0) and Q = (0, 0, 1). Use Eq. (8) to show that X lies on this plane if and only if y = z.
Derive a formula for the volume of the wedge in Figure 19(B) in terms of the constants a, b, and c. (B)
Use the formula for arc length to show that the length of a graph over [1, 4] cannot be less than 3.
Compute the Laplace transform Lƒ(s) of the function ƒ(x) = x2eαx for s > α.
Which of the following curves pass through the point (1, 4)?(a) c(t) = (t2, t + 3) (b) c(t) = (t2, t − 3)(c) c(t) = (t2, 3 − t)(d) c(t) = (t − 3, t2)
Points P and Q with the same radial coordinate (choose the correct answer):(a) Lie on the same circle with the center at the origin.(b) Lie on the same ray based at the origin.
Find parametric equations for the line through P = (2, 5) perpendicular to the line y = 4x − 3.
Find the speed (as a function of t) of a particle whose position at time t seconds is c(t) = (sin t + t, cos t + t). What is the particle’s maximal speed?
Calculate dy/dx at the point indicated.c(t) = (ln t, 3t2 − t), P = (0, 2)
Find the length of (3et − 3, 4et + 7) for 0 ≤ t ≤ 1.
Let c(t) = (e−t cos t, e−t sin t).Show that c(t) for 0 ≤ t < ∞ has finite length and calculate its value.
Let c(t) = (e−t cos t, e−t sin t).Find the first positive value of t0 such that the tangent line to c(t0) is vertical, and calculate the speed at t = t0.
Plot c(t) = (sin 2t, 2 cos t) for 0 ≤ t ≤ π. Express the length of the curve as a definite integral, and approximate it using a computer algebra system.
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