Question #116077
4.Suppose that α=2i−3j+k and \eta=7i−5j+k , find a unit vector perpendicular to α
and \(\eta) respectively.
\\(\\frac{-2i+5j+11k }{5\\sqrt(6)}\\)
\\(\\frac{2i-5j+11k }{5\\sqrt(6)}\\)
\\(\\frac{2i+5j-11k }{5\\sqrt(6)}\\)
\\(\\frac{2i+5j+11k }{5\\sqrt(6)}\\)
1
Expert's answer
2020-05-20T19:51:49-0400

α=2i3j+k,  β=7i5j+k\alpha =2i-3j+k, \ \ \beta =7i-5j+k

The cross product α×β\alpha \times \beta is a vector that is perpendicular to both α\alpha and  β\beta .

γ=α×β=ijk231751=(3×1(5)×1)i(2×17×1)j+(2×(5)7(3))k=2i+5j+11k\gamma =\alpha \times \beta =\begin{vmatrix} i&j&k \\ 2&-3&1\\7&-5&1 \end{vmatrix}=(-3\times 1-(-5)\times 1)i-(2\times 1-7\times 1)j+(2\times (-5)-7(-3))k=2i+5j+11k


γ=22+52+112=150=56|\gamma|=\sqrt{2^2+5^2+11^2}=\sqrt{150}=5\sqrt6

Unit vector is γγ\frac{\gamma}{|\gamma|} .


Answer: 2i+5j+11k56\frac{2i+5j+11k}{5\sqrt6}  a unit vector perpendicular to α\alpha and β\beta respectively. 



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