Find the angle between the vectors. (First find an exact expression and then approximate to the nearest degree.) a = 4, 3 , b = 2, −1
> 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.
> Find, correct to the nearest degree, the three angles of the triangle with the given vertices. P (2, 0), Q (0, 3), R (3, 4)
> Find the angle between the vectors. (First find an exact expression and then approximate to the nearest degree.) a = 8i - j + 4k, b = 4j + 2k
> Find a ∙ b. a = 5, −2 , b = 3, 4
> Find the angle between the vectors. (First find an exact expression and then approximate to the nearest degree.) a = 4i - 3j + k, b = 2i - k
> What is the angle between the given vector and the positive direction of the x-axis? i + 3 j
> Find the angle between the vectors. (First find an exact expression and then approximate to the nearest degree.) a = −1, 3, 4 , b = 5, 2, 1
> Find the angle between the vectors. (First find an exact expression and then approximate to the nearest degree.) a = 1, −4, 1 , b = 0, 2, −2
> Find the angle between the vectors. (First find an exact expression and then approximate to the nearest degree.) a = −2, 5 , b = 5, 12
> (a). Draw the vectors a = 3, 2 , b = 2, −1 , and c = 7, 1 . (b). Show, by means of a sketch, that there are scalars s and t such that c = sa + tb. (c). Use the sketch to estimate the values of s and t. (d). Find the exact values of s and t.
> Is the line through (-4, -6, 1) and (-2, 0, -3) parallel to the line through (10, 18, 4) and (5, 3, 14)?
> Is the line through (-2, 4, 0) and (1, 1, 1) perpendicular to the line through (2, 3, 4) and (3, -1, -8)?
> Describe in words the region of R3 represented by the equation(s) or inequality. x = z
> A woman walks due west on the deck of a ship at 3 mi/h. The ship is moving north at a speed of 22 mi/h. Find the speed and direction of the woman relative to the surface of the water.
> Describe in words the region of R3 represented by the equation(s) or inequality. x2 + y2 + z2 < 4
> Describe in words the region of R3 represented by the equation(s) or inequality. x2 + y2 + z2 = 4
> Find the magnitude of the resultant force and the angle it makes with the positive x-axis. y. 20 lb 45° 30° 16 lb
> Suppose that the cost (in dollars) for a company to produce x pairs of a new line of jeans is C(x) = 2000 + 3x + 0.01x2 + 0.0002x3 (a) Find the marginal cost function. (b) Find C’(100) and explain its meaning. What does it predict? (c) Compare C’(100) wi
> Describe in words the region of R3 represented by the equation(s) or inequality. y2 = 4
> Describe in words the region of R3 represented by the equation(s) or inequality. 0 < z < 6
> Describe in words the region of R3 represented by the equation(s) or inequality. z > - 1
> Describe in words the region of R3 represented by the equation(s) or inequality. y < 8
> Describe in words the region of R3 represented by the equation(s) or inequality. y = 22
> Describe in words the region of R3 represented by the equation(s) or inequality. x = 5
> Find a unit vector that has the same direction as the given vector. -5i + 3j - k
> Find a unit vector that has the same direction as the given vector. 6, −2
> Find an equation of a sphere if one of its diameters has endpoints (5, 4, 3) and (1, 6, -9).
> (a). Prove that the midpoint of the line segment from P1(x1, y1, z1) to P2(x2, y2, z2) is (b). Find the lengths of the medians of the triangle with vertices A (1, 2, 3), B (-2, 0, 5), and C (4, 1, 5). (A median of a triangle is a line segment that join
> Find dy/dx by implicit differentiation. xey = x - y
> Show that the equation represents a sphere, and find its center and radius. 3x2 + 3y2 + 3z2 = 10 + 6y + 12z
> Show that the equation represents a sphere, and find its center and radius. 2x2 + 2y2 + 2z2 = 8x - 24z + 1
> Find the sum of the given vectors and illustrate geometrically. 1, 3, −2 , 0, 0, 6
> Find the sum of the given vectors and illustrate geometrically. 3, 0, 1 , 0, 8, 0
> Find an equation of the sphere that passes through the origin and whose center is (1, 2, 3).
> Find an equation of the sphere that passes through the point (4, 3, -1) and has center (3, 8, 1).
> Refer to the law of laminar flow given in Example 7. Consider a blood vessel with radius 0.01 cm, length 3 cm, pressure difference 3000 dynes/cm2, and viscosity = 0.027. (a) Find the velocity of the blood along the center-liner = 0, at radius r = 0.005
> Find the lengths of the sides of the triangle PQR. Is it a right triangle? Is it an isosceles triangle? P (2, -1, 0), Q (4, 1, 1), R (4, -5, 4).
> If the vectors in the figure satisfy |u| − |v| = 1 and u + v + w = 0, what is |w|? u
> In the figure, the tip of c and the tail of d are both the midpoint of QR. Express c and d in terms of a and b. P b a R
> Copy the vectors in the figure and use them to draw the following vectors. (a). a + b (b). a - b (c). 1 2 a (d). -3b (e). a + 2b (f). 2b - a b a
> Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If f is continuous on [a, b], then f(9) dx – x [f(4) dx
> Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If f is continuous on [a, b], then 5f (x) dx = 5f(x) dx %3|
> Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If ƒ and g are continuous on [a, b], then r(x) dx g(x) dx
> Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If f and g are continuous on [a, b], then SS) + g(x)]dx = [S(4) dx + f* g(x) dx
> Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If f has a discontinuity at 0, then |. f(x) dx does not exist.
> Evaluate the indefinite integral. Illustrate and check that your answer is reasonable by graphing both the function and its antiderivative (take C = 0). x3 -dx x² + 1
> The table shows how the average age of first marriage of Japanese women has varied since 1950. (a) Use a graphing calculator or computer to model these data with a fourth-degree polynomial. (b) Use part (a) to find a model for A’(t).
> Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. 3 dx = 8 -2 X
> Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. So (x – x) dx represents the area under the curve y = x – x³ from 0 to 2.
> Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If f is continuous on [a, b], then d f(x) dx ) -f(x) dx
> Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If fo f(x) dx = 0, then f(x) = 0 for 0 <x< 1.
> Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. fe*ax = {f e*dx + f, e* dx ex* dx J5 Jo
> Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. All continuous functions have antiderivatives.
> Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. All continuous functions have derivatives.
> Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. L (ax? + bx + c) dx = 2 (ax? + c) dx
> Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. sin x – 6xº + dx = 0 - (1 + x*)?
> Evaluate the indefinite integral. Illustrate and check that your answer is reasonable by graphing both the function and its antiderivative (take C = 0). cs X dx V1 + sin x
> Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If f and g are differentiable and f(x) > g(x) for a <x < b, then f'(x) > g'(x) for a <x < b.
> The table gives the population of the world P(t), in millions, where t is measured in years and t = 0 corresponds to the year 1900. (a) Estimate the rate of population growth in 1920 and in 1980 by averaging the slopes of two secant lines. (b) Use a g
> Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If f and g are continuous and f(x) > g(x) for a < x< b, then g(x) dx
