10.8E: Vectors (Exercises)

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For the following exercises, determine whether the two vectors, $\mathbf{u}$ and $\mathbf{v},$ are equal, where $\mathbf{u}$ has an initial point $P_{1}$ and a terminal point $P_{2},$ and $\mathbf{v}$ has an initial point $P_{3}$ and a terminal point $P_{4}$.

52. $P_{1}=(-1,4), P_{2}=(3,1), P_{3}=(5,5)$ and $P_{4}=(9,2)$

53. $P_{1}=(6,11), P_{2}=(-2,8), P_{3}=(0,-1)$ and $P_{4}=(-8,2)$

For the following exercises, use the vectors $\mathbf{u}=2 \mathbf{i}-\mathbf{j}, \mathbf{v}=4 \mathbf{i}-3 \mathbf{j},$ and $w=-2 \mathbf{i}+5 \mathbf{j}$ 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. $\langle 6,-2\rangle$

59. $\langle-3,-3\rangle$

For the following exercises, calculate $\mathbf{u} \cdot \mathbf{v}$.

60. $u=-2 i+j$ and $v=3 i+7 j$

61. $u=i+4 j$ and $v=4 i+3 j$

62. Given $\boldsymbol{v}=\langle-3,4\rangle$ draw $\boldsymbol{v}, 2 \boldsymbol{v},$ and $\frac{1}{2} \boldsymbol{v}$

63. Given the vectors shown in Figure 4 , sketch $\boldsymbol{u}+\boldsymbol{v}, \boldsymbol{u}-\boldsymbol{v}$ and $3 \boldsymbol{v}$.

$Diagram of vectors v, 2v, and 1/2 v. The 2v vector is in the same direction as v but has twice the magnitude. The 1/2 v vector is in the same direction as v but has half the magnitude.$

Figure 4

64. Given initial point $P_{1}=(3,2)$ and terminal point $P_{2}=(-5,-1),$ write the vector $\mathbf{v}$ in terms of $\mathbf{i}$ and $\mathbf{j}$. Draw the points and the vector on the graph.