Question

If d subscript 1 baseline overscript right-arrow endscripts plus d subscript 2 baseline overscript right-arrow endscripts equals 7.0 d subscript 3 baseline overscript right-arrow endscripts, d subscript 1 baseline overscript right-arrow endscripts minus d subscript 2 baseline overscript right-arrow endscripts equals 1.0 d subscript 3 baseline overscript right-arrow endscripts and d overscript right-arrow endscripts subscript 3 baseline equals 2.1 i overscript ̂ endscripts plus 5.0 j overscript ̂ endscripts, then what are (a) the x component of d subscript 1 baseline overscript right-arrow endscripts, (b) the y component of d subscript 1 baseline overscript right-arrow endscripts, (c) the x component of d subscript 2 baseline overscript right-arrow endscripts, and (d) the y component of d subscript 2 baseline overscript right-arrow endscripts?

Answer 1

Given that


`\overrightarrow{d_1}+\overrightarrow{d_2}=5.0\overrightarrow{d_3}, \ and \\ \\ \overrightarrow{d_1}-\overrightarrow{d_2}=1.0\overrightarrow{d_3}`

Also, given that

`\overrightarrow{d_3}=2.6i+5.6j`

Let

`\overrightarrow{d_1}=x_1i+y_1j`, and

`\overrightarrow{d_2}=x_2i+y_2j`, then


`x_1i+y_1j+x_2i+y_2j=5.0(2.6i+5.6j) \\ \\ \Rightarrow(x_1+x_2)i+(y_1+y_2)j=13i+28j \\ \\ \Rightarrow x_1+x_2=13 \ . \ . \ . \ . \ . \ (1) \\ \\ \Rightarrow y_1+y_2=28 \ . \ . \ . \ . \ . \ (2)`

Also,


`(x_1i+y_1j)-(x_2i+y_2j)=1.0(2.6i+5.6j) \\ \\ \Rightarrow(x_1-x_2)i+(y_1-y_2)j=2.6i+5.6j \\ \\ \Rightarrow x_1-x_2=2.6 \ . \ . \ . \ . \ . \ (3) \\ \\ \Rightarrow y_1-y_2=5.6 \ . \ . \ . \ . \ . \ (4)`

Part A:

Solving for (1) and (3), we have


`x_1+x_2=13 \\ x_1-x_2=2.6 \\ \\ \Rightarrow2x_2=10.4 \\ \\ \Rightarrow x_2=5.2\\ \\ \Rightarrow x_1=7.8`

Therefore, the x-component of
`d_1` is 7.8



Part B

Solving for (2) and (4), we have


`y_1+y_2=28 \\ y_1-y_2=5.6 \\ \\ \Rightarrow2y_2=22.4 \\ \\ \Rightarrow y_2=11.2 \\ \\ \Rightarrow y_1=16.8`

Therefore, the y-component of
`d_1` is 16.8



Part C

Solving for (1) and (3), we have


`x_1+x_2=13 \\ x_1-x_2=2.6 \\ \\ \Rightarrow2x_2=10.4 \\ \\ \Rightarrow x_2=5.2\\ \\ \Rightarrow x_1=7.8`

Therefore, the x-component of
`d_2` is 5.2



Part D

Solving for (2) and (4), we have


`y_1+y_2=28 \\ y_1-y_2=5.6 \\ \\ \Rightarrow2y_2=22.4 \\ \\ \Rightarrow y_2=11.2 \\ \\ \Rightarrow y_1=16.8`

Therefore, the y-component of
`d_2` is 11.2