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Question
The vectors `vec"a" = 3hat"i" - 2hat"j" + 2hat"k"` and `vec"b" = -hat"i" - 2hat"k"` are the adjacent sides of a parallelogram. The acute angle between its diagonals is ______.
Solution
The vectors `vec"a" = 3hat"i" - 2hat"j" + 2hat"k"` and `vec"b" = -hat"i" - 2hat"k"` are the adjacent sides of a parallelogram. The acute angle between its diagonals is `pi/4`.
Explanation:
Given that `vec"a" = 3hat"i" - 2hat"j" + 2hat"k"`
And `vec"b" = -hat"i" - 2hat"k"`
∴ `vec"a" + vec"b" = 2hat"i" - 2hat"j"` and `vec"a" - vec"b" = 4hat"i" - 2hat"j" + 4hat"k"`
Let θ be the angle between the two diagonal vectors `vec"a" + vec"b"` and `vec"a" - vec"b"` then
`cos theta = ((vec"a" + vec"b") * (vec"a" - vec"b"))/(|vec"a" + vec"b"||vec"a" - vec"b"|)`
= `((2hat"i" - 2hat"j")*(4hat"i" - 2hat"j" + 4hat"k"))/(sqrt((2)^2 + (-2)^2) * sqrt((4)^2 + (-2)^2 + (4)^2)`
= `(8 + 4)/(2sqrt(2)*6)`
= `12/(2sqrt(2)*6)`
= `1/sqrt(2)`
∴ `theta = pi/4`
Hence the value of required filler is `pi/4`.
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