Course: 16.9 答复-第9章:矢量

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答复 - Ch 9:矢量
Section 9.2: Two-Dimensional Vectors
::第9.2节:两维矢量

Answers vary.
::答案不尽相同。

<-5, -9>

<5, 9>

<-2, -17>

<-3, 8>

<-8, 2>

<8, -2>

Magnitude: 7.62

::磁度: 7.62

Magnitude: 5

::磁度: 5

Magnitude: 11.18

::磁度: 11.18

Magnitude: 10

::磁度: 10

No
::否无

No
::否无

<-4.97, -12.01>

NE at 10 knots
::NN 10海里内

Pick a direction for the boat's movement (along one axis), then set the direction of the stream along the other. If, for example, you set the boat to move in the positive $\begin{aligned} x \end{aligned}$ direction and the stream to move in the negative $\begin{aligned} y \end{aligned}$ direction, then the vector would be <35, -25>.
::选择船只移动的方向( 沿着一个轴) , 然后沿着另一个轴设置溪流的方向。例如, 如果您设定船只向正向 x 方向移动, 则向量向负 y 方向移动, 则向量为 < 35, - 25 > 。

Section 9.3: Operations with Vectors
::第9.3节:向量操作

$\begin{aligned} < 9, 7 > \end{aligned}$

$\begin{aligned} < - 10, 10 > \end{aligned}$

$\begin{aligned} < 4, 22 > \end{aligned}$

$\begin{aligned} < 15, - 15 > \end{aligned}$

$\begin{aligned} 11.4 \end{aligned}$

$\begin{aligned} 14.14 \end{aligned}$

$\begin{aligned} 22.36 \end{aligned}$

21.21

$\begin{aligned} 53.13 \end{aligned}$ °

$\begin{aligned} 116.57 \end{aligned}$ °

$\begin{aligned} 306.87 \end{aligned}$ °

$\begin{aligned} 17 \end{aligned}$ miles
::17英里

The bird is actually moving in a direction of $\begin{aligned} 28 \end{aligned}$ ° south of east ( $\begin{aligned} 62 \end{aligned}$ ° east of south).
::这只鸟实际上正在向东向南28度方向移动(向南62度)。

$\begin{aligned} 50.6 \end{aligned}$ miles per hour
::每小时50.6英里/小时

$\begin{aligned} 9 \end{aligned}$ ° east of north
::北部以东9度

Section 9.4: Unit Vectors
::第9.4节:单位矢量

$\begin{aligned} y = 18 \end{aligned}$
::y=18岁

$\begin{aligned} < \frac{2}{\sqrt{29}}, \frac{5}{\sqrt{29}} > \end{aligned}$

$\begin{aligned} < \frac{9}{\sqrt{82}}, - \frac{1}{\sqrt{82}} > \end{aligned}$

$\begin{aligned} < \frac{1}{\sqrt{10}}, - \frac{3}{\sqrt{10}} > \end{aligned}$

$\begin{aligned} < \frac{1}{\sqrt{82}}, \frac{9}{\sqrt{82}} > \end{aligned}$

(-0.32, 0.66)

(-0.37, 0.68)

(1.47, -0.24)

$\begin{aligned} < 37.1, 29.8 > \end{aligned}$

$\begin{aligned} < 200, 200 \sqrt{3} > \end{aligned}$

$\begin{aligned} < 10.26, 28.19 > \end{aligned}$

$\begin{aligned} < - 153.9, 422.86 > \end{aligned}$

$\begin{aligned} < - 423.5, - 79.9 > \end{aligned}$

374.2 mph at 263.14°
::263.14°时为374.2毫米/小时

16.9 mph north and 36.25 mph east
::北纬16.9米,东经36.25米

Section 9.5: Dot Products
::第9.5节:点产品

$\begin{aligned} 24 \end{aligned}$

$\begin{aligned} 16 \end{aligned}$

$\begin{aligned} - 14 \end{aligned}$

$\begin{aligned} 21 \end{aligned}$

$\begin{aligned} 34 \end{aligned}$

$\begin{aligned} 49.4 \end{aligned}$ °

$\begin{aligned} 56.3 \end{aligned}$ °

$\begin{aligned} 171.87 \end{aligned}$ °

$\begin{aligned} 8.1 \end{aligned}$ °

$\begin{aligned} 26.6 \end{aligned}$ °

$\begin{aligned} {v_{1}}^{2} + {v_{2}}^{2} \end{aligned}$
::v12+v22 (v12+v22)

The magnitude of a vector is the square root of the dot product of the vector with itself.
::矢量的大小是矢量本身的点产物的平方根。

0

$\begin{aligned} (c u) \cdot v & =< c u_{1} > \cdot < v_{1}, v_{2} > \\ =< c u_{1} v_{1} + c u_{2} v_{2} > \\ =< u_{1} \cdot c v_{1} + u_{2} \cdot c v_{2} > \\ =< u_{1}, u_{2} > \cdot < c v_{1}, c v_{2} > \\ = u \cdot (c v) \end{aligned}$
:1111222u1cv1222221212222222222222

To prove that these two vectors are perpendicular to each other, show that their dot products are equal to zero.
::证明这两个矢量相互垂直,显示其点值产品等于零。

53.46 J
::53.46 J

142.6 J
::142.6.6 J

246.2 J
::246.2 J 246.2 J

Section 9.6: Scalar and Vector Projections
::第9.6节: 斯卡弧和矢量预测

$\begin{aligned} < 6, 114 > \end{aligned}$

$\begin{aligned} < - 56, - 64 > \end{aligned}$

$\begin{aligned} < - 105, 45 > \end{aligned}$

$\begin{aligned} < 99, - 220 > \end{aligned}$

$\begin{aligned} < 144, 160 > \end{aligned}$

$\begin{aligned} {p r o j}_{A} \end{aligned}$ → $\begin{aligned} B = \end{aligned}$ $\begin{aligned} < \frac{27}{17}, \frac{45}{17} > \end{aligned}$
::B=<2717,4517>

$\begin{aligned} p r o j \end{aligned}$ → $\begin{aligned} = \end{aligned}$ < $\begin{aligned} \frac{16}{5}, \frac{48}{5} \end{aligned}$ >
::proj = < 165,485 >

$\begin{aligned} {p r o j}_{c} \end{aligned}$ → $\begin{aligned} D = \end{aligned}$ < $\begin{aligned} \frac{23}{122}, \frac{253}{122} \end{aligned}$ >
::D=<23122,253122>

$\begin{aligned} {p r o j}_{E} \end{aligned}$ → $\begin{aligned} F = \end{aligned}$ < $\begin{aligned} - \frac{59}{325}, \frac{1062}{325} \end{aligned}$ >
::F= <-59325,1062325 >

$\begin{aligned} p r o j \end{aligned}$ → $\begin{aligned} = \end{aligned}$ < $\begin{aligned} - \frac{80}{61}, \frac{96}{61} \end{aligned}$ >
::= <-8061,9661>

$\begin{aligned} {p r o j}_{H} \end{aligned}$ → $\begin{aligned} I = \end{aligned}$ < $\begin{aligned} - \frac{298}{229}, \frac{2235}{229} \end{aligned}$ >
::= < 298229,22352229>

$\begin{aligned} p r o j \end{aligned}$   → $\begin{aligned} = \end{aligned}$ < $\begin{aligned} \frac{11}{5}, \frac{22}{5} \end{aligned}$ >
::= < 115 225>

$\begin{aligned} p r o j \end{aligned}$ → $\begin{aligned} = \end{aligned}$ < $\begin{aligned} \frac{399}{113}, \frac{456}{113} \end{aligned}$ >
::= < 399113456113>

$\begin{aligned} {p r o j}_{J} \end{aligned}$ → $\begin{aligned} K = \end{aligned}$ < $\begin{aligned} - \frac{224}{41}, \frac{280}{41} \end{aligned}$ >
::K= <-22441,28041>

$\begin{aligned} {p r o j}_{L} \end{aligned}$ → $\begin{aligned} M = \end{aligned}$ < $\begin{aligned} \frac{161}{338}, \frac{391}{338} \end{aligned}$ >
::M=<161338,391338>

34.64

62.5

29.34



Section 9.7: Vector Equation of a Line
::第9.7节:线条矢量等量

$\begin{aligned} \vec{r} = \end{aligned}$ $\begin{aligned} ⟨ 2, - 2 ⟩ \end{aligned}$ $\begin{aligned} + k \end{aligned}$ $\begin{aligned} ⟨ - 1, 7 ⟩ \end{aligned}$
::2,2,2 +k +1,7

$\begin{aligned} \vec{r} = \end{aligned}$ $\begin{aligned} ⟨ 2, - 9 ⟩ \end{aligned}$ $\begin{aligned} + k \end{aligned}$ $\begin{aligned} ⟨ 6, 13 ⟩ \end{aligned}$
::+k 6,13 +2,9,9,9 +k +6,13

$\begin{aligned} \vec{r} = \end{aligned}$ $\begin{aligned} ⟨ 15, 3 ⟩ \end{aligned}$ $\begin{aligned} + k \end{aligned}$ $\begin{aligned} ⟨ - 11, - 6 ⟩ \end{aligned}$
::============================================================================================================================================== ===================================================================================== ========================================================================================================================================================================================================================================================================================

$\begin{aligned} \vec{r} = \end{aligned}$    $\begin{aligned} ⟨ - 1, - 1 ⟩ \end{aligned}$ $\begin{aligned} + k \end{aligned}$ $\begin{aligned} ⟨ 4, 12 ⟩ \end{aligned}$
::+k 4,12

$\begin{aligned} \vec{r} = \end{aligned}$ $\begin{aligned} ⟨ 1, - 3 ⟩ \end{aligned}$ $\begin{aligned} + k \end{aligned}$ $\begin{aligned} ⟨ - 6, 6 ⟩ \end{aligned}$
::1,3,3 +k,6,6

$\begin{aligned} \vec{r} = \end{aligned}$ $\begin{aligned} ⟨ 25, 17 ⟩ \end{aligned}$ $\begin{aligned} + k \end{aligned}$ < $\begin{aligned} ⟨ 41, 5 ⟩ \end{aligned}$ >
::25 17 +k 41 5 +k 41 5 +k

Skew
::扭曲扭曲

Skew
::扭曲扭曲

Skew
::扭曲扭曲

Skew
::扭曲扭曲

(1.5, 1.5)

(5, -2.5)

(9.5, 0)

(1, 5)

$\begin{aligned} t = 3 : \end{aligned}$ $\begin{aligned} ⟨ 9, 16 ⟩ \end{aligned}$ ; $\begin{aligned} t = 5 : \end{aligned}$ $\begin{aligned} ⟨ 13, 30 ⟩ \end{aligned}$
::t=3:$9,16;t=5:$13,30

$\begin{aligned} t = 3 : \end{aligned}$ $\begin{aligned} ⟨ 28, - 25 ⟩ \end{aligned}$ ; $\begin{aligned} t = 5 : \end{aligned}$ $\begin{aligned} ⟨ 46, - 39 ⟩ \end{aligned}$
::t=3: 28,-25 ; t=5: 46,-39

$\begin{aligned} t = 3 : \end{aligned}$    $\begin{aligned} ⟨ 8, - 1 ⟩ \end{aligned}$ ; $\begin{aligned} t = 5 : \end{aligned}$ $\begin{aligned} ⟨ 14, - 3 ⟩ \end{aligned}$
::t=3:8-1;t=5:14,3

16.9 答复-第9章:矢量

Section outline

General

答复 - Ch 9:矢量

Section 9.2: Two-Dimensional Vectors
::第9.2节:两维矢量

Section 9.3: Operations with Vectors
::第9.3节:向量操作

Section 9.4: Unit Vectors
::第9.4节:单位矢量

Section 9.5: Dot Products
::第9.5节:点产品

Section 9.6: Scalar and Vector Projections
::第9.6节: 斯卡弧和矢量预测

Section 9.7: Vector Equation of a Line
::第9.7节:线条矢量等量

Section outline

General

答复 - Ch 9:矢量

Section 9.2: Two-Dimensional Vectors ::第9.2节:两维矢量

Section 9.3: Operations with Vectors ::第9.3节:向量操作

Section 9.4: Unit Vectors ::第9.4节:单位矢量

Section 9.5: Dot Products ::第9.5节:点产品

Section 9.6: Scalar and Vector Projections ::第9.6节: 斯卡弧和矢量预测

Section 9.7: Vector Equation of a Line ::第9.7节:线条矢量等量

Section 9.2: Two-Dimensional Vectors
::第9.2节:两维矢量

Section 9.3: Operations with Vectors
::第9.3节:向量操作

Section 9.4: Unit Vectors
::第9.4节:单位矢量

Section 9.5: Dot Products
::第9.5节:点产品

Section 9.6: Scalar and Vector Projections
::第9.6节: 斯卡弧和矢量预测

Section 9.7: Vector Equation of a Line
::第9.7节:线条矢量等量