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Find two unit vectors each of which is perpendicular to both uu¯ and vv¯ where uijku¯=2i^+j^-2k^, vjv¯=j-2^k′. - Mathematics and Statistics

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Sum

Find two unit vectors each of which is perpendicular to both `bar"u"` and `bar"v"` where `bar"u" = 2hat"i" + hat"j" - 2hat"k"`,  `bar"v" = hat"i" + 2hat"j" - 2hat"k"`.

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Solution

Let `bar"u" = 2hat"i" + hat"j" - 2hat"k"`, 
`bar"v" = hat"i" + 2hat"j" - 2hat"k"`

Then `bar"u" xx bar"v" = |(hat"i",hat"j",hat"k"),(2,1,-2),(1,2,-2)|`

`= (- 2 + 4)hat"i" + (- 4 + 2)hat"j" + (4 - 1)hat"k"`

`= 2hat"i" - 2hat"j" + 3hat"k"`

`∴ |bar"u" xx bar"v"| = sqrt((2)^2 + (-2)^2 + (3)^2)`

`= sqrt(4 + 4 + 9)`

`= sqrt(17)`

`= +-(bar"u" xx bar"v")/(|bar"u" xx bar"v"|) = +-( 2hat"i" - 2hat"j" + 3hat"k")/sqrt17`

`= +-(2/sqrt17hat"i" - 2/sqrt17hat"j" + 3/sqrt17hat"k")`

Concept: Vector Product of Vectors (Cross)
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