# If aijka¯=2i^+j^-3k^ and bijkb¯=i^-2j^+k^, find a vector of magnitude 5 perpendicular to both aa¯ and bb¯. - Mathematics and Statistics

Sum

If bar"a" = 2hat"i" + hat"j" - 3hat"k" and  bar"b" = hat"i" - 2hat"j" + hat"k", find a vector of magnitude 5 perpendicular to both bar"a" and bar"b".

#### Solution

Given: bar"a" = 2hat"i" + hat"j" - 3hat"k" and

bar"b" = hat"i" - 2hat"j" + hat"k"

∴ bar"a" xx bar"b" = |(hat"i", hat"j", hat"k"),(2,1,-3),(1,-2,1)|

= (1 - 6)hat"i" - (2 + 3)hat"j" + (- 4 - 1)hat"k"

= - 5hat"i" - 5hat"j" - 5hat"k"

∴ |bar"a" xx bar"b"| = sqrt((-5)^2 + (- 5)^2 + (- 5)^2)

= sqrt(25 + 25 +25) = sqrt75 = 5sqrt3

∴ unit vectors perpendicular to both the vectors bar"a" and bar"b"

= (+- (bar"a"xxbar"b"))/(|bar"a" xx bar"b"|)

= (+- (- 5hat"i" - 5hat"j" - 5hat"k"))/(5sqrt3)

= (+- (hat"i" + hat"j" + hat"k"))/sqrt3

∴ required vectors of magnitude 5 units

= +- 5/sqrt3 (hat"i" + hat"j" + hat"k").

#### Notes

[Note: Answer in the textbook is incorrect.]

Concept: Vector Product of Vectors (Cross)
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