10.8E: Vectors (Exercises)

For the following exercises, determine whether the two vectors, 𝐮 and 𝐯, are equal, where 𝐮 has an initial point 𝑃1 and a terminal point 𝑃2, and 𝐯 has an initial point 𝑃3 and a terminal point 𝑃4.

52. 𝑃1 =(−1,4),𝑃2 =(3,1),𝑃3 =(5,5) and 𝑃4 =(9,2)

53. 𝑃1 =(6,11),𝑃2 =(−2,8),𝑃3 =(0,−1) and 𝑃4 =(−8,2)

For the following exercises, use the vectors 𝐮 =2⁢𝐢 −𝐣,𝐯 =4⁢𝐢 −3⁢𝐣, and 𝑤 =−2⁢𝐢 +5⁢𝐣 to evaluate the expression.

54. 𝑢 −𝑣

55. 2⁢𝑣 −𝑢 +𝑤

For the following exercises, find a unit vector in the same direction as the given vector.

56. 𝑎 =8⁢𝑖 −6⁢𝑗

57. 𝑏 =−3⁢𝑖 −𝑗

For the following exercises, find the magnitude and direction of the vector.

58. ⟨6,−2⟩

59. ⟨−3,−3⟩

For the following exercises, calculate 𝐮 ⋅𝐯.

60. 𝑢 =−2⁢𝑖 +𝑗 and 𝑣 =3⁢𝑖 +7⁢𝑗

61. 𝑢 =𝑖 +4⁢𝑗 and 𝑣 =4⁢𝑖 +3⁢𝑗

62. Given 𝒗 =⟨−3,4⟩ draw 𝒗,2⁢𝒗, and 12⁢𝒗

63. Given the vectors shown in Figure 4 , sketch 𝒖 +𝒗,𝒖 −𝒗 and 3⁢𝒗.

Figure 4

64. Given initial point 𝑃1 =(3,2) and terminal point 𝑃2 =(−5,−1), write the vector 𝐯 in terms of 𝐢 and 𝐣. Draw the points and the vector on the graph.