Problems 37 - 39: Solve the problem. Assume projectile is ideal, launch angle is measured from the horizontal, and launch is over a horizontal surface, unless stated otherwise. 24) An ideal projectile is launched from level ground at a launch angle of 26° and an initial speed of 48 m/sec. How far away from the launch point does the projectile hit the ground? A) ≈ 60 m B) ≈ 230 m C) ≈ 290 m D) ≈ 185 m 25) A projectile is fired with an initial speed of 528 m/sec at an angle of 45°. What is the greatest height reached by the projectile? Round answer to the nearest tenth of a meter. A) 7111.8 m B) 76.2 m C) 69,696.0 m D) 28,447.3 m Find the unit tangent vector of the given curve. 26) r(t) = (5 + 2t7 )i + (4 + 10t7 )j + (8 + 11t7 )k A) T(t) = C) T(t) = 2 2 11 i+ j+ k 15 3 15 2 2 11 i+ j+ k 225 45 225 B) T (t) = 14 14 77 i+ j+ k 15 3 15
D) 111.1°
C) 1.57
D) 1.79
B) Yes
C)
147 49 147 ij+ k 19 19 19
D)
58 58 319 i+ j+ k 15 3 15
C) - 45, 13, -36
D) 0
B) - 5/ 251, - 15/ 251, - 1/ 251 D) - 5, 15, 1
Calculate a and find the direction angle for the following vector. Give the direction angle as an angle in [0°, 360°) rounded to the nearest tenth. 11) a = - 1, 5 A) a = 26, 78.7° B) a = 26, 168.7° C) a = 26, 101.3° D) a = 26, 11.3°
D) r = - 16, - 4
C) 1, -16, - 2
D) 5, - 29, - 6
B) 74
C)
74
D) 12
4) Let v = 8 i + 6j - 9k and w = 2 i - 2j - 8k. Find v · w A) - 3 B) 12, 18, - 64
C) 76
D) - 48, 54, 72
Find a unit vector in the direction θ. 12) θ = 60° 1 1 A) 2, 2 2 2
B) -
1 1 , 2 2
3
C)
1 1 , 2 2
3
D)
1 2
3,
1 2
Solve the problem. 13) A force of magnitude 13 pounds pulling on a suitcase makes an angle of 30° with the ground. Express the force in terms of its i and j components. A) 6.500i + 11.26 j B) 11.26i + 6.500 j C) 0.8660i + + 0.5000j D) 2.005i - 12.84 j Find parametric equations for the line described below. 14) The line through the points P(- 1, - 1, - 7) and Q(3, - 6, 3) A) - 4 + t, 5 + t, - 10 - 7t C) - 1 + 4t, - 1 - 5t, - 7 + 10t
D)
13 15
For the smooth curve r(t), find the parametric equations for the line that is tangent to r at the given parameter value t = t0 . 19) r(t) = (2t2 - 3t)i + (t + 7)j + k ; t0 = 2 A) x = 2 + 5t, y = 9 + t, z = 1 C) x = 2 + 5t, y = 9 + t, z = 0 Evaluate the integral. 3 10t 4 20) i - 3t2 j + k dt 1+t (1 + t2 )2 0 9 9 A) 4 i - 27j + k B) 8i + 27j + k 2 10 B) x = 2 + t, y = t, z = t D) x = 5t, y = t, z = t
B) - t, - t, - 7t D) 1 + 4t, 1 - 5t, 7 + 10t
Find symmetric equne through P = ( - 4, 8, 3) and in the direction of v = - 7, - 4, 8 x- 7 y- 4 z+ 8 x+ 4 y-8 z- 3 A) B) = = = = 8 3 8 -4 -7 -4 C) x- 4 y+ 8 z+ 3 = = 8 -7 -4 D) x+ 7 y+4 z- 8 = = 8 3 -4
Find the angle between the given vectors in radians or degrees, as marked. 5) 3, - 3 , 4, 9 ; degrees A) 121.1° B) 55.6° C) 45.6° Find the angle between u and v in radians. 6) u = 7 i - 9 j - 10k, v = 2 i + 10j - 4 k A) 1.67 B) -0.22 Determine whether the vectors are perpendicular. 7) u = 6, 5 , v = - 7, 3 A) No Find the projection of u onto a unit vector in the direction of v. 8) u = 2 i + 10j + 11k, v = 3i - j + 3k 87 29 87 58 58 319 A) ij+ k B) i+ j+ k 19 19 19 225 45 225 Find the cross product r x v for the given vectors. 9) r = 3, - 9, - 7 , v = - 1, - 9, - 2 A) - 45, - 13, - 36 B) 45, - 13, 36 Find the indicated perpendicular vector. 10) A unit vector perpendicular to both 3, 1, 0 and - 1, 0, - 5 A) 5, - 15, -1 C) - 5/ 251, 15/ 251, 1/ 251
MATH 2415 - Sample Problems for the Final Exam Find a position vector r that is equivalent to the vector PQ defined by points P and Q. 1) P(- 7, - 6) and Q(- 9, 2) A) r = 2, - 8 B) r = - 13, 2 C) r = - 2, 8 Find the following. 2) Let u = 7, - 3, 6 , v = - 9, 2, -6 , and w = - 3, 9, 2 . Find 2v + w - 3u. A) - 4, 23, 12 B) -42, 22, - 28 3) Let w = 8 i + 1j + 3k. Find the length of - w. A) 12
Write an equation for the plane determined by the given conditions. 16) Normal vector n = 4, 5, 7 and containing the point P = (1, 0, - 7) A) 4x + 5y + 7z - 45 = 0 B) x - 7z + 45 = 0 C) 4x + 5y + 7z + 45 = 0 D) x + y + z + 6 = 0 17) Containing the points P = (3, 1, 3), Q = (3, 0, - 3), and R = (0, 2, - 1) A) - y + 6z + 19 = 0 B) 10x - 18y - 3z + 3 = 0 C) 10x + 18y - 3z = 0 D) 10x + 18y - 3z - 39 = 0 Find the perpendicular distance from the given point to the given plane. 18) P = ( -10, 3, - 3); 2x + 11y + 10z = - 4 13 13 13 A) B) C) 225 5 75