Construct A and L matrices. A= 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 L= 1 , 164 , 499.387 − 4 , 660 , 600.879 4 , 182 , 158.213 1 , 164 , 499.400 − 4 , 660 , 600.884 4 , 182 , 158.214 1 , 168 , 947.849 − 4 , 650 , 766.255 4 , 191 , 851.271 1 , 168 , 947.841 − 4 , 650 , 766.249 4 , 191 , 851.263 Given that Baseline vector are given as: Explanation: Writing observation equation for X coordinate for bonnieto ray: X B − R = X B + Δ X B − R + v 1 WHERE X B = X coordinate of the station bonnie = 1 , 161 , 121.599 m ΔX B - R = X component in baseline vector = 3.377.788 m ʋ 1 = residual X B − R = 1 , 161 , 121.599 + 3.377.788 + v 1 = 1 , 164 , 499.387 + v 1 Writing observation equation for Y coordinate for Bonnie to Ray: Y B − R = Y B + Δ Y B − R + v 2 W H E R E Y B = Y c o o r d i n a t e o f t h e s t a t i o n B O N N I E = − 4 , 655 , 872.977 Δ Y B − R = Y c o m p o n e n t i n b a s e l i n e v e c t o r = − 4 , 727.902 v 2 = r e s i d u a l Y B − R = − 4 , 655 , 872.977 + − 4 , 727.902 + v 2 = − 4 , 660 , 600.879 + v 2 Writing observation equation for Z coordinate for Bonnie to Ray: Z B − R = X B + Δ Z B − R + v 3 W H E R E Z B = Z c o o r d i n a t e o f t h e s t a t i o n b o n n i e = 4 , 188 , 330.232 m Δ Z B − R = Z c o m p o n e n t i n b a s e l i n e v e c t o r = − 6 , 172.019 m v 3 = r e s i d u a l Z J − T = 4 , 188 , 330.232 − 6 , 172.019 + v 3 = 4 , 182 , 158.213 + v 3 Writing observation equation for X coordinate for tom to herb: X T − H = X T + Δ X T − H + v 4 W H E R E X T = X c o o r d i n a t e o f t h e s t a t i o n T o m = 1 , 176 , 398.558 m Δ X T − H = X c o m p o n e n t i n b a s e l i n e v e c t o r = − 7 , 450.717 m v 4 = r e s i d u a l X T − H = 1 , 176 , 398.558 − 7 , 450.717 + v 4 = 1 , 168 , 947.841 + v 4 Writing observation equation for y coordinate for tom to herb: Y T − H = Y T + Δ Y T − H + v 5 W H E R E Y T = Y c o o r d i n a t e o f t h e s t a t i o n t o m = − 4 , 653 , 039.613 m Δ Y T − H = Y c o m p o n e n t i n b a s e l i n e v e c t o r = 2 , 273.364 m v 5 = r e s i d u a l Y A − T = − 4 , 653 , 039.613 + 2 , 273.364 + v 5 = − 4 , 650 , 766.249 + v 5 Writing observation equation for Z coordinate for tom to herb Z T − H = Z T + Δ Z T − H + v 6 W H E R E Z T = Z c o o r d i n a t e o f t h e s t a t i o n t o m = 4 , 187 , 198.360 m Δ Z T − H = Z c o m p o n e n t i n b a s e l i n e v e c t o r = 4652.903 m v 6 = r e s i d u a l Z T − H = 4 , 187 , 198.360 m + 4652.903 + v 6 = 4 , 191 , 851.263 + v 6 Writing observation equations for X coordinate for bonnie to herb: X B − H = X B + Δ X B − H + v 7 W H E R E X B = X c o o r d i n a t e o f t h e s t a t i o n T o m = 1 , 161 , 121.599 m Δ X B − H = X c o m p o n e n t i n b a s e l i n e v e c t o r = 7826.248 m v 7 = r e s i d u a l X B − H = 1 , 161 , 121.599 + 7826.248 + v 7 = 1 , 168 , 947.847 + v 7 Writing observation equation for Y coordinates for bonnie to herb: Y B - H = Y B + ΔY B - H +ʋ 8 WHERE Y B = Y coordinate of the station Tom = -4,655,872.977m ΔY B - H = Y component in baseline vector = 5,106.722m ʋ 8 = residual X A - T =4,655,872.977+5,106.722+ʋ 8 = -4,650,766.255+ ʋ 8 Writing observation equation for Z coordinates for bonnie to herb: Z B − H = Z B + Δ Z B − H + v 9 W h e r e , Z B = Z c o o r d i n a t e o f t h e s t a t i o n T o m = 4 , 188 , 330.232 m Δ Z B − H = Z c o m p o n e n t i n b a s e l i n e v e c t o r = 3 , 521.039 m v 9 = r e s i d u a l Z B − H = 4 , 188 , 330.232 + 3 , 521.039 + v 9 = 4 , 191 , 851.271 + v 9 Writing observation equation for X coordinate for tom to ray: X T − R = X T + Δ X T − R + v 10 W H E R E X T = X c o o r d i n a t e o f t h e s t a t i o n T o m = 1 , 176 , 398.558 m Δ X T − R = X c o m p o n e n t i n b a s e l i n e v e c t o r = − 11 , 899.158 m v 10 = r e s i d u a l X T − R = 1 , 176 , 398.558 − 11 , 899.15 + v 10 = 1 , 164 , 499.400 + v 10 Writing observation equation for Y coordinate for tom to ray: Y T − R = Y T + Δ Y T − R + v 11 W H E R E Y T = Y c o o r d i n a t e o f t h e s t a t i o n T o m = − 4 , 653 , 039.613 m Δ Y T − R = Y c o m p o n e n t i n b a s e l i n e v e c t o r = − 7 , 561.271 m v 11 = r e s i d u a l Y T − R = 1 , 176 , 398.558 − 11 , 899.15 + v 11 = − 4 , 660 , 600.884 + v 11 Writing observation equation for Z coordinate for tom to ray: Z T − R = Z T + Δ Z T − R + v 12 W H E R E Z T = Y c o o r d i n a t e o f t h e s t a t i o n T o m = 4 , 187 , 198.360 m Δ Z T − R = Y c o m p o n e n t i n b a s e l i n e v e c t o r = − 5 , 040.146 m v 12 = r e s i d u a l Z A − T = 4 , 187 , 198.360 − 5 , 040.146 + v 12 = − 4 , 182 , 158.214 + v 12 A = X B − R 0 0 0 0 0 0 Y B − R 0 0 0 0 0 0 Z B − R 0 0 0 0 0 0 X T − R 0 0 0 0 0 0 Y T − R 0 0 0 0 0 0 Z T − R X B − H 0 0 0 0 0 0 Y B − H 0 0 0 0 0 0 Z B − H 0 0 0 0 0 0 X T − H 0 0 0 0 0 0 Y T − H 0 0 0 0 0 0 Z T − H A= 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 Construct L matrix: L = 1 , 164 , 499.387 − 4 , 660 , 600.879 4 , 182 , 158.213 1 , 164 , 499.400 − 4 , 660 , 600.884 4 , 182 , 158.214 1 , 168 , 947.849 − 4 , 650 , 766.255 4 , 191 , 851.271 1 , 168 , 947.841 − 4 , 650 , 766.249 4 , 191 , 851.263 Conclusion: A= 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 L= 1 , 164 , 499.387 − 4 , 660 , 600.879 4 , 182 , 158.213 1 , 164 , 499.400 − 4 , 660 , 600.884 4 , 182 , 158.214 1 , 168 , 947.849 − 4 , 650 , 766.255 4 , 191 , 851.271 1 , 168 , 947.841 − 4 , 650 , 766.249 4 , 191 , 851.263

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Question

Chapter 16, Problem 16.25P

To determine

Construct A and L matrices.

A= 100000010000001000000100000010000001100000010000001000000100000010000001

L= 1,164,499.387−4,660,600.8794,182,158.2131,164,499.400−4,660,600.8844,182,158.2141,168,947.849−4,650,766.2554,191,851.2711,168,947.841−4,650,766.2494,191,851.263

Given that

Baseline vector are given as:

Explanation:

Writing observation equation for X coordinate for bonnieto ray:

XB−R= XB+ ΔXB−R+v1

WHERE

X_B= X coordinate of the station bonnie = 1,161,121.599m

ΔX_B_-_R= X component in baseline vector = 3.377.788m

ʋ₁= residual

XB−R=1,161,121.599+3.377.788+v1          = 1,164,499.387+ v1

Writing observation equation for Y coordinate for Bonnie to Ray:

YB−R= YB+ ΔYB−R+v2WHEREYB= Y coordinate of the station BONNIE = −4,655,872.977ΔYB−R= Y component in baseline vector = −4,727.902v2= residual YB−R=−4,655,872.977+−4,727.902+v2          = −4,660,600.879+ v2

Writing observation equation for Z coordinate for Bonnie to Ray:

ZB−R= XB+ ΔZB−R+v3WHEREZB= Z coordinate of the station bonnie = 4,188,330.232mΔZB−R= Z component in baseline vector = −6,172.019mv3= residual ZJ−T=4,188,330.232−6,172.019+v3          = 4,182,158.213+ v3

Writing observation equation for X coordinate for tom to herb:

XT−H= XT+ ΔXT−H+v4WHEREXT= X coordinate of the station Tom = 1,176,398.558mΔXT−H= X component in baseline vector = −7,450.717mv4= residual XT−H=1,176,398.558 −7,450.717+v4          = 1,168,947.841+ v4

Writing observation equation for y coordinate for tom to herb:

YT−H= YT+ ΔYT−H+v5WHEREYT= Y coordinate of the station tom = −4,653,039.613mΔYT−H= Y component in baseline vector = 2,273.364mv5= residual YA−T=−4,653,039.613+2,273.364+ v5          = −4,650,766.249+ v5

Writing observation equation for Z coordinate for tom to herb

ZT−H= ZT+ ΔZT−H+v6WHEREZT= Z coordinate of the station tom = 4,187,198.360mΔZT−H= Z component in baseline vector = 4652.903mv6= residual ZT−H=4,187,198.360m +4652.903+v6          = 4,191,851.263+ v6

Writing observation equations for X coordinate for bonnie to herb:

XB−H= XB+ ΔXB−H+v7WHEREXB= X coordinate of the station Tom = 1,161,121.599mΔXB−H= X component in baseline vector = 7826.248mv7= residual XB−H=1,161,121.599+7826.248+v7          = 1,168,947.847+ v7

Writing observation equation for Y coordinates for bonnie to herb:

Y_B_-_H= Y_B+ ΔY_B_-_H+ʋ₈WHERE

Y_B= Y coordinate of the station Tom = -4,655,872.977m

ΔY_B_-_H= Y component in baseline vector = 5,106.722m

ʋ₈= residual

X_A_-_T=4,655,872.977+5,106.722+ʋ₈ = -4,650,766.255+ ʋ₈

Writing observation equation for Z coordinates for bonnie to herb:

ZB−H= ZB+ ΔZB−H+v9Where,ZB= Z coordinate of the station Tom = 4,188,330.232mΔZB−H= Z component in baseline vector = 3,521.039m v9= residual ZB−H=4,188,330.232+3,521.039+v9          = 4,191,851.271+ v9

Writing observation equation for X coordinate for tom to ray:

XT−R= XT+ ΔXT−R+v10WHEREXT= X coordinate of the station Tom = 1,176,398.558mΔXT−R= X component in baseline vector = −11,899.158mv10= residual XT−R=1,176,398.558 −11,899.15+v10          = 1,164,499.400+ v10

Writing observation equation for Y coordinate for tom to ray:

YT−R= YT+ ΔYT−R+v11WHEREYT= Y coordinate of the station Tom = −4,653,039.613mΔYT−R= Y component in baseline vector = −7,561.271mv11= residualYT−R=1,176,398.558 −11,899.15+v11          = −4,660,600.884+ v11

Writing observation equation for Z coordinate for tom to ray:

ZT−R= ZT+ ΔZT−R+v12WHEREZT= Y coordinate of the station Tom = 4,187,198.360mΔZT−R= Y component in baseline vector = −5,040.146mv12= residualZA−T=4,187,198.360−5,040.146+v12= −4,182,158.214+ v12

A = XB−R000000YB−R000000ZB−R000000XT−R000000YT−R000000ZT−RXB−H000000YB−H000000ZB−H000000XT−H000000YT−H000000ZT−H

A= 100000010000001000000100000010000001100000010000001000000100000010000001

Construct L matrix:

L = 1,164,499.387−4,660,600.8794,182,158.2131,164,499.400−4,660,600.8844,182,158.2141,168,947.849−4,650,766.2554,191,851.2711,168,947.841−4,650,766.2494,191,851.263

Conclusion:

A= 100000010000001000000100000010000001100000010000001000000100000010000001

L= 1,164,499.387−4,660,600.8794,182,158.2131,164,499.400−4,660,600.8844,182,158.2141,168,947.849−4,650,766.2554,191,851.2711,168,947.841−4,650,766.2494,191,851.263

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