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Question
Find all vectors of magnitude `10sqrt(3)` that are perpendicular to the plane of `hat"i" + 2hat"j" + hat"k"` and `-hat"i" + 3hat"j" + 4hat"k"`
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Solution
Let `vec"a" = hat"i" + 2hat"j" + hat"k"` and `vec"b" = -hat"i" + 3hat"j" + 4hat"k"`.
Then `vec"a" xx vec"b" = |(hat"i", hat"j", hat"k"),(1, 2, 1),(-1, 3, 4)|`
= `hat"i"(8 - 3) - hat"j"(4 + 1) + hat"k"(3 + 2)`
= `5hat"i" - 5hat"j" + 5hat"k"`
⇒ `|vec"a" xx vec"b"| = sqrt((5)^2 + (-5)^2 + (5)^2)`
= `sqrt(3(5)^2)`
= `5sqrt(3)`
Therefore, unit vector perpendicular to the plane of `vec"a"` and `vec"b"` is given by
`(vec"a" xx vec"b")/|vec"a" xx vec"b"| = (5hat"i" - 5hat"j" + 5hat"k")/(5sqrt(3)`
Hence, vectors of magnitude of `10sqrt(3)` that are perpendicular to plane of `vec"a"` and `vec"b"` are `+-10sqrt(3) ((5hat"i" - 5hat"j" + 5hat"k")/(5sqrt(3)))`
i.e., `+- 10(hat"i" - hat"j" + hat"k")`.
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