smc-1177-10-Cube/Rect prism

`ABCDEFGH` are the vertices of a rectangular prism.

Show that the internal diagonals of the prism, `AG` and `DF`, intersect. (2 marks)

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Calculate the acute angle, `theta`, between the diagonals, to the nearest minute. (2 marks)

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i.	`A(2, text{−2}, 0),`	`G(text{−2}, 2, 2)`
	`D(2, 2, 0),`	`F (text{−2}, text{−2}, 2)`

`text(Midpoint)\ AG = ((1/2 (2-2)),(1/2 (text{−2} + 2)),(1/2 (0 + 2))) = ((0), (0), (1))`

`text(Midpoint)\ DF = ((1/2 (2-2)),(1/2 (2-2)),(1/2 (0 + 2))) = ((0), (0), (1))`

`text(S) text(ince midpoints are the same), AG and DF\ text(intersect.)`

ii.	`vec(AG) = ((text{−2}), (2), (2))-((2), (text{−2}), (0)) = ((text{−4}), (4), (2))`
	`vec(DF) = ((text{−2}), (text{−2}), (2))-((2), (2), (0)) = ((text{−4}), (text{−4}), (2))`

`vec (AG) ⋅ vec (DF) = |\ vec (AG)\ | ⋅ |\ vec(DF)\ |\ cos theta`

`((text{−4}), (4), (2)) ⋅ ((text{−4}), (text{−4}), (2)) = sqrt 36 sqrt 36 cos theta`

A cube with side length 3 units is pictured below.

Calculate the magnitude of vector `vec(AG)`. (1 mark)

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Find the acute angle between the diagonals `vec(AG)` and `vec(BH)`. (3 marks)

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i. `A(3, 0 , 0), \ \ G(0, 3, 3)`

ii.	`H (3, 3, 3)`
	`vec(BH) = ((3), (3), (3))`

Vectors, SPEC2 SM-Bank 22