A point P is on the rim of a moving bicycle wheel of radius 1 ft. The bicycle is moving

Calculate the vectors in Exercises 18, using a = (5, 1, 0), b = (2, 1, 9), c = (1, 1, 2). 1.3c 2

Calculate the vectors in Exercises 18, using a = (5, 1, 0), b = (2, 1, 9), c = (1, 1, 2). 1. a +b

Calculate the vectors in Exercises 18, using a = (5, 1, 0), b = (2, 1, 9), c = (1, 1, 2). 1. b a 4.

Calculate the vectors in Exercises 18, using a = (5, 1, 0), b = (2, 1, 9), c = (1, 1, 2). 1. a + 2(b + c ) 5.

Calculate the vectors in Exercises 18, using a = (5, 1, 0), b = (2, 1, 9), c = (1, 1, 2). 1. 2a 3(b c ) 6.

Calculate the vectors in Exercises 18, using a = (5, 1, 0), b = (2, 1, 9), c = (1, 1, 2). 1. 2(b + 4(a + c )) 7.

Calculate the vectors in Exercises 18, using a = (5, 1, 0), b = (2, 1, 9), c = (1, 1, 2). 1. a +b (a b ) 8. 4

Calculate the vectors in Exercises 18, using a = (5, 1, 0), b = (2, 1, 9), c = (1, 1, 2). 1. 4(c + 2(a c ) 2a ) For

For Exercises 916, perform the indicated computations. 4(i 2j ) 0.5(i k ) 10.

For Exercises 916, perform the indicated computations. (i + 2j )+(3)(2i +j ) 11.

For Exercises 916, perform the indicated computations. (3i +j ) (5i 2j ) 12.

For Exercises 916, perform the indicated computations. (3j 2 k +i ) (4 k 2i + 3j ) 13. (5

For Exercises 916, perform the indicated computations. (5i j 3 k ) (2i +j + k ) 14. (

For Exercises 916, perform the indicated computations. (2i 5j )(i +j + k ) 15. (

For Exercises 916, perform the indicated computations. (2i + 5j ) 3i (i +j + k ) 16. (i

For Exercises 916, perform the indicated computations. (i +j + k )(2i + 3j + k ) (3i +j + 4 k ) Resolve t

Resolve the vectors in Exercises 1718 into components.

Resolve the vectors in Exercises 1718 into components.

Resolve the vectors in Exercises 1718 into components.

Resolve the vectors in Exercises 1718 into components.

(a) Are the vectors 4i +aj +6 k and ai +(a1)j +3 k parallel for any values of the constant a? (b) Are these vectors ever perpendicular? 22.

A point P is on the rim of a moving bicycle wheel of radius 1 ft. The bicycle is moving forward at 6 ft/sec. (a) Sketch the velocity of P relative to the wheels axle at the 3, 6, 9, 12 oclock positions. (b) Sketch the velocity of the axle relative to the ground. (c) Sketch the velocity of P relative to the ground at the positions in part (a). (d) Does P ever stop, relative to the ground? What is the greatest speed that P moves, relative to the ground?

A gymnastics academy offers classes to children of different ages. Software used by the academy tracks enrollment data using vectors. Let E = (e1, e2, e3, e4, e5) represent the number of children enrolled at each of five different experience levels, from beginner (1) to advanced (5), and let E max = (40, 40, 30, 15, 10) be the maximum allowed enrollment at each level. (a) Let E = (51, 47, 41, 22, 23) give the number of applicants at each level for the next session. After full enrollment is reached, the remaining applicants are placed on a wait list, L . Find L . (b) At the start of the next session, E = E max. Let D = (8, 4, 9, 7, 6) be the number of dropouts after the first week of the next session. Find F = E D and G = L D . What do these vectors tell you about enrollment? 24. Let E be t

Let E be the enrollment vector for the gymnastics academy described in Problem 23, where E max = (40, 40, 30, 15, 10). Also, let T = (30, 30, 40, 80, 120) be the weekly tuition (in $). (a) Evaluate E max T . What does this tell you about the gymnastics academy? (b) Let R = (5, 5, 10, 20, 30) describe a rate hike in weekly tuition, and let T new = T + R . Evaluate E max R and E max T new. What do these two quantities tell you about tuition? In Problems

In Problems 2527, a 5-pound block sits on a plank of wood. If one end of the plank is raised, the block slides down the plank. However, friction between the block and the plank prevents it from sliding until the plank has been raised a certain height. It turns out that the sliding force exerted by gravity on the block is proportional to the sine of the angle made by the plank with the ground. Find a formula for F = f(), the sliding force (in lbs) exerted on a block if the plank makes an angle of with the ground. [Hint: What is sliding force if the plank is horizontal? Vertical?]

In Problems 2527, a 5-pound block sits on a plank of wood. If one end of the plank is raised, the block slides down the plank. However, friction between the block and the plank prevents it from sliding until the plank has been raised a certain height. It turns out that the sliding force exerted by gravity on the block is proportional to the sine of the angle made by the plank with the ground. One end of the plank is lifted at a constant rate of 2 ft per second, while the other end rests on the ground. (a) Find a formula for F = h(t), the sliding force exerted on the block as a function of time. (b) Suppose the block begins to slide once the sliding force equals 3 lbs. At what time will the block begin to slide?

The 5-lb force exerted on the block by gravity can be resolved into two components, the sliding force FS parallel to the ramp and the normal force F N perpendicular to the ramp. Use this information to show that your formula in Problem 25 is correct. 28

A plane is heading due east and climbing at the rate of 80 km/hr. If its airspeed is 480 km/hr and there is a wind blowing 100 km/hr to the northeast, what is the ground speed of the plane?

A particle moving with speed v hits a barrier at an angle of 60 and bounces off at an angle of 60 in the opposite direction with speed reduced by 20 percent, as shown in Figure 12.34. Find the velocity vector of the object after impact.

Figure 12.35 shows a molecule with four atoms at O, A, B and C. Show that every atom in the molecule is 2 units away from every other atom.

Two cylindrical cans of radius 2 and height 7 are shown in Figure 12.36. The cans touch down the side. Let A be the point at the top rims where they touch. Let B be the front point on the bottom rim of the left can, and C be the back point on the bottom rim of the right can. The origin is at A and the z-axis points upward; the x-axis points forward (out of the paper) and the y-axis points to the right. (a) Write vectors in component form for AB and AC. (b) What is the angle between AB and AC?

(a) Using the fact that u v = ||u || ||v || cos , show that u (v ) = (u v ). [Hint: What happens to the angle when you multiply v by 1?] (b) Using the fact that u v = ||u || ||v || cos , show that for any negative scalar u (v ) = (u v ) (u ) v = (u v ). 33. A consumption v

A consumption vector of three goods is given by x = (x1, x2, x3), where x1, x2 and x3 are the quantities consumed of the three goods. Consider a budget constraint represented by the equation p x = k, where p is the price vector of the three goods and k is a constant. Show that the difference between two consumption vectors corresponding to points satisfying the same budget constraint is perpendicular to the price vector p . 34.

Consider the grid in Figure 12.37. Write expressions for AB and CD in terms ofi and j . A

Consider the grid of equilateral triangles in Figure 12.38. Find expressions for AB, BC, AC, and AD in terms of u and v . A

Consider the regular hexagon in Figure 12.39. Express the six sides and all three diameters in terms of m and n . m

Consider the grid of regular hexagons in Figure 12.40. Express AC, AB, AD and DB in terms of m and n.