Q27E - University Physics with Modern Physics

Question

Compute the x- and y-components of the vectors $\overrightarrow{A},\overrightarrow{B},\overrightarrow{C} \mathbf{and} \overrightarrow{D}$ in Fig. E1.24.

                                                  

Step-by-Step Solution

Answer

The x and the y components of vector  A, B, C and D are  (0,-8) ; (7.5 , 13); (-10.88,-5.07) and (-7.99, 6.02)

respectively ,

Step 1: Representation of vector

Here, vector  is  easily defined with the help of its  magnitude and its  angle with the x and the y axis.  

Here in this case, there are four given vectors. 

For this vector we will use the given  magnitude   as well as the angle with the x axis and the y axis.

Step 2: Conversion of mile into kilometer

For the direction shown in the graph, 

$\overrightarrow{A}$ =(0, -8)  in meters

Here, the vector is exactly on the y axis so, the magnitude with respect to y axis will be zero.

 

For vector B, the angle of the  vector made with the x axis will be  90 – 30, 

that  is the angle made will be 60. 

Hence, the components of the B  vector will be  magnitude multiplied by the cos angle and sin angle made with respect to x axis. 

 

$\overrightarrow{B}=15\cos 60+15 \sin 60$

By putting the value of cos 60 = ½ and sin 60 = $\sqrt{3}$/2 

We get, 

$\overrightarrow{B}$ = (7.5 , 13) in meters

 

Similarly for vector C 

$\overrightarrow{C}=-12\cos 25-12\sin 25$ 

After the putting the value of cos 25 and sin 25, 

 

C = (-10.88,-5.07) in meters

The coordinates for vector D will be, 

 

$\overrightarrow{D}=-10\cos 53+10\sin 53$

 D = (-7.99, 6.02) in meter

 

Hence, the coordinates of A, B, C and D are  (0,-8) ; (7.5 , 13); (-10.88,-5.07) and (-7.99, 6.02), respectively.