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Question
Find a vector of magnitude 6, which is perpendicular to both the vectors `2hat"i" - hat"j" + 2hat"k"` and `4hat"i" - hat"j" + 3hat"k"`.
Solution
Let `2hat"i" - hat"j" + 2hat"k"` and `4hat"i" - hat"j" + 3hat"k"`
We know that unit vector perpendicular to `vec"a"` and `vec"b" = ((vec"a" xx vec"b"))/|vec"a" xx vec"b"|`
`vec"a" xx vec"b" = |(hat"i", hat"j", hat"k"),(2, -1, 2),(4, -1, 3)|`
= `hat"i"(-3 + 2) - hat"j"(6 - 8) + hat"k"(-2 + 4)`
= `-hat"i" + 2hat"j" + 2hat"k"`
∴ `|vec"a" xx vec"b"| = sqrt((-1)^2 + (2)^2 + (2)^2)`
= `sqrt(1 + 4 + 4)`
= `sqrt(9)`
= 3
So, `((vec"a" xx vec"b"))/|vec"a" xx vec"b"| = (-"i" + 2hat"j" + 2hat"k")/3`
= `1/3(-hat"i" + 2hat"j" + 2hat"k")`
Now the vector of magnitude 6 = `1/3(-hat"i" + 2hat"j" + 2hat"k") * 6`
= `2(-hat"i" + 2hat"j" + 2hat"k")`
= `-2hat"i" + 4hat"j" + 4hat"k"`
Hence, the required vector is `-2hat"i" + 4hat"j" + 4hat"k"`.
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