(a) Find a nonzero vector orthogonal to the plane through the points P, Q, and R, and (b) find the area of triangle PQR. P (2, -3, 4), Q (-1, -2, 2), R (3, 1, -3)
> Prove Formula 6 of Theorem 3.
> Prove Formula 3 of Theorem 3.
> Prove Formula 1 of Theorem 3.
> Evaluate the integral. t 1 k) dt 1 - t j+ V1 – 12
> Evaluate the integral. (sec?t i + t(t? + 1)°j + t² In t k) dt
> Evaluate the integral. /
> Sketch the curve with the given vector equation. Indicate with an arrow the direction in which t increases. r(t) = t2 i + t4 j + t6 k
> Evaluate the integral. 1 i + t + 1 j+ t2 + 1 k dt 1² + 1
> Evaluate the integral. S (22 i+ (t + 1)/F k) đt 3/2
> Evaluate the integral. S (ti - r'j+ 3r°k) dt Jo
> Use Theorem 10 to find the curvature. r(t) = 6 t2 i + 2t j + 2t3 k
> Use Theorem 10 to find the curvature. r(t) = t i + t2 j + et k
> Use Theorem 10 to find the curvature. r(t) = t3 j + t2 k
> (a). Find the unit tangent and unit normal vectors T(t) and N(t). (b). Use Formula 9 to find the curvature. r(t) = (r, }r°, r²)
> Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. Illustrate by graphing both the curve and the tangent line on a common screen. x = 2 cos t, y = 2 sin t, z = 4 cos 2t; ( 3 , 1, 2)
> Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. Illustrate by graphing both the curve and the tangent line on a common screen. x = t, y = e-t, z = 2t - t2; (0, 1, 0)
> Sketch the curve with the given vector equation. Indicate with an arrow the direction in which t increases. r(t) = 2 cos t i + 2 sin t j + k
> (a). Find the unit tangent and unit normal vectors T(t) and N(t). (b). Use Formula 9 to find the curvature. r(t) = (/2t, e', e-)
> Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. /t2 + 3, y= ln(t² + 3), z = t; (2, In 4, 1) X =
> Suppose you start at the point (0, 0, 3) and move 5 units along the curve x = 3 sin t, y = 4t, z = 3 cos t in the positive direction. Where are you now?
> Find the unit tangent vector T (t) at the point with the given value of the parameter t. r (t) = cos t i + 3t j + 2 sin 2t k, t = 0
> Let C be the curve of intersection of the parabolic cylinder x2 = 2y and the surface 3z = xy. Find the exact length of C from the origin to the point (6, 18, 36).
> Find the unit tangent vector T (t) at the point with the given value of the parameter t. r(t) = (r² – 21, 1 + 3t, + ), t= 2
> Find the derivative of the vector function. r (t) = t a × (b + t c)
> Find the derivative of the vector function. r (t) = a + t b + t2 c
> Find the length of the curve. r(t) = t2 i + 9t j + 4t3/2 k, 1 < t < 4
> Find the derivative of the vector function. 1 r(t) 1 +t -i + 1 + t j + k 1 + t
> Find the derivative of the vector function. r (t) = t2 i + cos (t2) j + sin2t k
> Find the derivative of the vector function. r(t) = (Vi – 2,3, 1/r²)
> The figure shows a curve C given by a vector function r (t). (a). Draw the vectors r (4.5) – r (4) and r (4.2) – r (4). (b). Draw the vectors (c). Write expressions for r' (4) and the unit tangent vector T (4). (d).
> Find parametric equations and symmetric equations for the line. The line through (-6, 2, 3) and parallel to the line 1 2 x – 1 3 y = z + 1
> If a = i - 2k and b = j + k, find a × b. Sketch a, b, and a × b as vectors starting at the origin.
> Find the cross product a × b and verify that it is orthogonal to both a and b. a = t i + cos t j + sin tk, b = i - sin t j + cos tk
> Find the cross product a × b and verify that it is orthogonal to both a and b. a = 1 2 i + 1 3 j + 1 4 k, b = i + 2 j – 3k
> Show that |a × b |2 = |a |2 | b |2 - (a ∙ b)2.
> (a). Let P be a point not on the line L that passes through the points Q and R. Show that the distance d from the point P to the line L is (b). Use the formula in part (a) to find the distance from the point P (1, 1, 1) to the line through Q (0, 6, 8)
> (a). Find all vectors v such that 1, 2, 1 × v = 3, 1, −5 (b). Explain why there is no vector v such that 1, 2, 1 × v = 3, 1, 5
> Find the unit vectors that are parallel to the tangent line to the parabola y − x2 at the point (2, 4).
> If a ∙ b = 3 and a × b = 1, 2, 2 , find the angle between a and b.
> A wrench 30 cm long lies along the positive y-axis and grips a bolt at the origin. A force is applied in the direction 0, 3, −4 at the end of the wrench. Find the magnitude of the force needed to supply 100 N ∙ m of torque to the bolt.
> Find the cross product a × b and verify that it is orthogonal to both a and b. a = 3i + 3j - 3k, b = 3i - 3j + 3k
> A particle moves according to a law of motion s = f(t), t ≥ 0, where t is measured in seconds and s in feet. (a) Find the velocity at time t. (b) What is the velocity after 1 second? (c) When is the particle at rest? (d) When is the part
> Use the scalar triple product to determine whether the points A (1, 3, 2), B (3, -1, 6), C (5, 2, 0), and D (3, 6, -4) lie in the same plane.
> Use the scalar triple product to verify that the vectors u = i + 5 j - 2 k, v = 3i - j, and w = 5i + 9 j - 4 k are coplanar.
> Find the volume of the parallelepiped with adjacent edges PQ, PR, and PS. P (3, 0, 1), Q (-1, 2, 5), R (5, 1, -1), S (0, 4, 2)
> Find the volume of the parallelepiped with adjacent edges PQ, PR, and PS. P (-2, 1, 0), Q (2, 3, 2), R (1, 4, -1), S (3, 6, 1)
> Find the volume of the parallelepiped determined by the vectors a, b, and c. a = i + j, b = j + k, c = i + j + k
> Find the volume of the parallelepiped determined by the vectors a, b, and c. a = 1, 2, 3 , b = −1, 1, 2 , c = 2, 1, 4
> A crane suspends a 500-lb steel beam horizontally by support cables (with negligible weight) attached from a hook to each end of the beam. The support cables each make an angle of 60° with the beam. Find the tension vector in each support cabl
> (a) Find a nonzero vector orthogonal to the plane through the points P, Q, and R, and (b) find the area of triangle PQR. P (0, -2, 0), Q (4, 1, -2), R (5, 3, 1)
> (a) Find a nonzero vector orthogonal to the plane through the points P, Q, and R, and (b) find the area of triangle PQR. P (0, 0, -3), Q (4, 2, 0), R (3, 3, 1)
> Find the cross product a × b and verify that it is orthogonal to both a and b. a = 2j - 4k, b = 2i + 3j + k
> Find dy/dx by implicit differentiation. y cos x = x2 + y2
> Suppose y = f (x) is a curve that always lies above the x-axis and never has a horizontal tangent, where f is differentiable everywhere. For what value of y is the rate of change of y5 with respect to x eighty times the rate of change of y with respect t
> Find the area of the parallelogram with vertices P (1, 0, 2), Q (3, 3, 3d), R (7, 5, 8), and S (5, 2, 7).
> Prove the property of cross products (Theorem 11). Property 3: a × (b + c) = a × b + a × c
> Prove the property of cross products (Theorem 11). Property 2: (ca) × b = c (a × b) = a × (cb)
> If p(x) is the total value of the production when there are x workers in a plant, then the average productivity of the workforce at the plant is A(x) = p(x) / x (a) Find A’(x). Why does the company want to hire more workers if A’(x) > 0? (b) Show that A’
> Prove the property of cross products (Theorem 11). Property 1: a × b = -b × a
> Show that (a × b) ∙ b = 0 for all vectors a and b in V3.
> Show that 0 × a = 0 = a × 0 for any vector a in V3.
> Find two unit vectors orthogonal to both j - k and i + j.
> Find the cross product a × b and verify that it is orthogonal to both a and b. a = 4, 3, −2 , b = 2, −1, 1
> Find two unit vectors orthogonal to both 3, 2, 1 and −1, 1, 0 .
> If a = 1, 0, 1 , b = 2, 1, −1 , and c = 0, 1, 3 , show that a × (b × c) ≠ (a × b) × c.
> If a = 2, −1, 3 and b = 4, 2, 1 , find a × b and b × a.
> Find |u × v | and determine whether u × v is directed into the page or out of the page. 5. |v|=16 120° l미=12
> Find the magnitude of the resultant force and the angle it makes with the positive x-axis. y. 200 N 300 N 60°
> Find a ∙ b. |a | = 7, |b | = 4, the angle between a and b is 30°
> Find a ∙ b. a = 3i + 2j - k, b = 4i + 5k
> Find a ∙ b. a = 2i + j, b = i - j + k
> Show that if u + v and u - v are orthogonal, then the vectors u and v must have the same length.
> The Parallelogram Law states that (a). Give a geometric interpretation of the Parallelogram Law. (b). Prove the Parallelogram Law. | a + b]° + ]a – b|² = 2|a |² + 2| b |?
> The Triangle Inequality for vectors is (a). Give a geometric interpretation of the Triangle Inequality. (b). Use the Cauchy-Schwarz Inequality from Exercise 61 to prove the Triangle Inequality. [Hint: Use the fact that |a + b |2 = (a + b) âˆ
> Prove Properties 2, 4, and 5 of the dot product (Theorem 2).
> The gas law for an ideal gas at absolute temperature T (in kelvins), pressure P (in atmospheres), and volume V (in liters) is PV = nRT, where n is the number of moles of the gas and R = 0.0821 is the gas constant. Suppose that, at a certain instant, P =
> If c = |a |b + |b |a, where a, b, and c are all nonzero vectors, show that c bisects the angle between a and b.
> If a child pulls a sled through the snow on a level path with a force of 50 N exerted at an angle of 380 above the horizontal, find the horizontal and vertical components of the force.
> A boat sails south with the help of a wind blowing in the direction S36°E with magnitude 400 lb. Find the work done by the wind as the boat moves 120 ft.
> A sled is pulled along a level path through snow by a rope. A 30-lb force acting at an angle of 40° above the horizontal moves the sled 80 ft. Find the work done by the force.
> A tow truck drags a stalled car along a road. The chain makes an angle of 30° with the road and the tension in the chain is 1500 N. How much work is done by the truck in pulling the car 1 km?
> Find a ∙ b. a = 4, 1, 1 4 , b = 6, −3, −8
> Suppose that a and b are nonzero vectors. (a). Under what circumstances is compa b − compb a? (b). Under what circumstances is proja b − projb a?
> For the vectors in Exercise 40, find ortha b and illustrate by drawing the vectors a, b, proja b, and ortha b. Exercise 40: Find the scalar and vector projections of b onto a. a = 1, 4 , b = 2, 3
> Show that the vector ortha b = b - proja b is orthogonal to a. (It is called an orthogonal projection of b.)
> Find the scalar and vector projections of b onto a. a = i + 2j + 3k, b = 5i - k
> Find the scalar and vector projections of b onto a. a = 3i - 3j + k, b = 2i + 4j - k
> Invasive species often display a wave of advance as they colonize new areas. Mathematical models based on random dispersal and reproduction have demonstrated that the speed with which such waves move is given by the function f (r) = Dr , where r is the
> Find the scalar and vector projections of b onto a. a = −1, 4, 8 , b = 12, 1, 2
> Find the scalar and vector projections of b onto a. a = 4, 7, −4 , b = 3, −1, 1
> Find the scalar and vector projections of b onto a. a = 1, 4 , b = 2, 3
> Find a ∙ b. a = 6, −2, 3 , b = 2, 5, −1
> Find the scalar and vector projections of b onto a. a = −5, 12 , b = 4, 6
> Find the direction cosines and direction angles of the vector. (Give the direction angles correct to the nearest degree.) 1 2 i + j + k
> Find the acute angles between the curves at their points of intersection. (The angle between two curves is the angle between their tangent lines at the point of intersection.) y = x2, y = x3
> What is the angle between the given vector and the positive direction of the x-axis? 8i + 6j
> Find a ∙ b. a = 1.5, 0.4 , b = −4, 6
> Find two unit vectors that make an angle of 608 with v = 3, 4 .
> Find a unit vector that is orthogonal to both i + j and i + k.
> Use vectors to decide whether the triangle with vertices P (1, -3, -2), Q (2, 0, -4), and R (6, -2, -5) is right-angled.
