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Question
Find the vectors of magnitude `10sqrt(3)` that are perpendicular to the plane which contains `hat"i" + 2hat"j" + hat"k"` and `hat"i" + 3hat"j" + 4hat"k"`
Solution
Let the given vectors be `vec"a" = hat"i" + 2hat"j" + hat"k"`
`vec"b" = hat"i" + 3hat"j" + 4hat"k"`
`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)`
`vec"a" xx vec"b" = 5hat"i" - 3hat"j" + hat"k"`
`|vec"a" xx vec"b"| = |5hat"i" - 3hat"j" + hat"k"|`
= `sqrt(5^2 + (-3)^2 + 1^2)`
`|vec"a" xx vec"b"| = sqrt(25 + 9 + 1)`
= `sqrt(35)`
The unit vector perpendicular to both `vec"a"` and `vec"b"`
= `+- (vec"a" xx vec"b")/|vec"a" xx vec"b"|`
= `+- (5hat"i" - 3hat"j" + hat"k")/sqrt(35)`
∴ The vector of magnitude `10sqrt(3)` perpendicular to `vec"a"` and `vec"b"`
= `+- 10sqrt(3) ((5hat"i" - 3hat"j" + hat"k")/sqrt(35))`
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