Question: Copy the vectors in the figure and
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
Transcribed Image Text:
b a
> 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
> Find the angle between the vectors. (First find an exact expression and then approximate to the nearest degree.) a = 4, 3 , b = 2, −1
> (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
> 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
> 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 [1, 3], then f'(v) dv = f(3) – f(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. If f is continuous on [a, b] and f(x) > 0, then L VF) dx f(x) dx
> Use the given graph of f to find the Riemann sum with six subintervals. Take the sample points to be (a) left endpoints and (b) midpoints. In each case draw a diagram and explain what the Riemann sum represents. y = f(x) 2- 6. 2.
> Boyle’s Law states that when a sample of gas is compressed at a constant temperature, the product of the pressure and the volume remains constant: PV = C. (a) Find the rate of change of volume with respect to pressure. (b) A sample of gas is in a contain
> Estimate the value of the number c such that the area under the curve y = sinh cx between x = 0 and x = 1 is equal to 1.
> A population of honeybees increased at a rate of r(t) bees per week, where the graph of r is as shown. Use the Midpoint Rule with six subintervals to estimate the increase in the bee population during the first 24 weeks. 12000 8000 4000 4 8 12 16 20
> In Example 6 we considered a bacteria population that doubles every hour. Suppose that another population of bacteria triples every hour and starts with 400 bacteria. Find an expression for the number n of bacteria after t hours and use it to estimate th
> A radar gun was used to record the speed of a runner at the times given in the table. Use the Midpoint Rule to estimate the distance the runner covered during those 5 seconds. 1 (s) v (m/s) I (s) v (m/s) 3.0 10.51 0.5 4.67 3.5 10.67 1.0 7.34 4.0 10.
> A particle moves along a line with velocity function v(t) = t2 - t, where v is measured in meters per second. Find (a) the displacement and (b) the distance traveled by the particle during the time interval [0, 5].
> Find the vector that has the same direction as 6, 2, −3 but has length 4.
> Find a unit vector that has the same direction as the given vector. 8i - j + 4k
> Find a + b, 4a + 2b, |a |, and |a - b |. a = 8, 1, −4 , b = 5, −2, 1
> Find a + b, 4a + 2b, |a |, and |a - b |. a = 4i - 3j + 2k, b = 2i - 4k
> Use vectors to prove that the line joining the midpoints of two sides of a triangle is parallel to the third side and half its length.
> The cost function for a certain commodity is C(q) = 84 + 0.16q - 0.0006q2 + 0.000003q3 (a) Find and interpret C9s100d. (b) Compare C’(100) with the cost of producing the 101st item.
> Prove Property 5 of vectors algebraically for the case n = 3. Then use similar triangles to give a geometric proof.
> Figure 16 gives a geometric demonstration of Property 2 of vectors. Use components to give an algebraic proof of this fact for the case n = 2.
> Find a + b, 4a + 2b, |a |, and |a - b |. a = 5i + 3j, b = -i - 2j
> What is the relationship between the point (4, 7) and the vector (4, 7)? Illustrate with a sketch.
> Find a + b, 4a + 2b, |a |, and |a - b |. a = −3, 4 , b = 9, −1
> Find an equation of the set of all points equidistant from the points A (-1, 5, 3) and B (6, 2, -2). Describe the set.
> Consider the points P such that the distance from P to A (-1, 5, 3d is twice the distance from P to B (6, 2, -2). Show that the set of all such points is a sphere, and find its center and radius.
> If A, B, and C are the vertices of a triangle, find АВ + ВС + СА
> Write inequalities to describe the region. The region consisting of all points between (but not on) the spheres of radius r and R centered at the origin, where r < R
> Differentiate. f (t) = cot t / et
> Find the sum of the given vectors and illustrate geometrically. 3, −1 , −1, 5
> Write inequalities to describe the region. The region between the yz-plane and the vertical plane x = 5
> Describe in words the region of R3 represented by the equation(s) or inequality. x2 + y2 + z2 > 2z
> Find the sum of the given vectors and illustrate geometrically. −1, 4 , 6, −2
> Find dy/dx by implicit differentiation. x2 / x + y = y2 + 1
> Find the local and absolute extreme values of the function on the given interval. f (x) = x3 - 9x2 + 24x - 2, [0, 5]
