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Solar Panels have to be installed carefully so that the tilt of the roof, and the direction to the sun, produce the largest possible electrical power in the solar panels.

A surveyor uses his instrument to determine the coordinates of the four corners of a roof where solar panels are to be mounted. In the picture , suppose the points are labelled counter clockwise from the roof corner nearest to the camera in units of meters P
_{
1
}
(6, 8, 4) , P
_{
2
}
(21, 8, 4), P
_{
3
}
(21, 16, 10) and P
_{
4
}
(6, 16, 10)

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Question 1
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What are the components to the two edge vectors defined by A ⃗ = PV of P
_{
2
}
– PV of P1 and B ⃗ = PV of P
_{
4
}
– PV of P
_{
1
}
? (where PV stands for position vector)

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Question 2
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## Write the vector in standard notation with i ̂, j ̂ and k ̂ (where i ̂, j ̂ and k ̂ are the unit vectors along the three axes).

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Question 3
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## What are the magnitudes of the vectors A ⃗ and B ⃗ and in what units?

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Question 4
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## What are the components to the vector N ⃗ , perpendicular to A ⃗ and B ⃗ and the surface of the roof?

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Question 5
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What is the magnitude of N ⃗ and its units? The sun is located along the unit vector S ⃗ = 1/2 i ̂-6/7 j ̂+1/7 k ̂. If the flow of solar energy is given by the vector F ⃗ = 910 S in units of watts/meter2 , what is the dot product of vectors F ⃗ 𝑤𝑖𝑡ℎ N ⃗ , and the units for this quantity?

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Question 6
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## What is the angle between vectors N ⃗ and S ⃗ ? What is the elevation angle of the sun above the plane of the roof? (cos 51° = 0.629 )