How do you find a unit vector that is orthogonal to a and b where $a = - 7 i + 6 j - 8 k$ $a = −7 i + 6 j − 8 k$ and $b = - 5 i + 6 j - 8 k$ $b = −5 i + 6 j − 8 k$ ?

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Answer 1 · 2016-12-28T08:56:47

The answer is $= \frac{1}{5} ⟨ 0, - 4, - 3 ⟩$ $=1/5〈0,-4,-3〉$

To find a vector orthogonal to 2 other vectors, we must do a cross product.

The cross product of 2 vectors, $\to a = ⟨ a, b, c ⟩$ $veca=〈a,b,c〉$ and $\to b = ⟨ d, e, f ⟩$ $vecb=〈d,e,f〉$

is given by the determinant

$| (hati,hatj,hatk), (a,b,c), (d,e,f) |$

and $| (a,b), (c,d) |=ad-bc$

Here, the 2 vectors are $veca=〈-7,6,-8〉$ and $〈-5,6,-8〉$

And the cross product is

$| (hati,hatj,hatk), (-7,6,-8), (-5,6,-8) |$

$=hati| (6,-8), (6,-8) | - hatj| (-7,-8), (-5,-8) |+hatk | (-7,6), (-5,6) |$

$=hati(-48+48)-hati(56-40)+hatk(-42+30)$

$=〈0,-16,-12〉$

Verification, by doing the dot product

$〈0,-16,-12〉.〈-7,6,-8〉=0-96+96=0$

$〈0,-16,-12〉.〈-5,6,-8〉=0-96+96=0$

Therefore, the vector is perpendicular to the other 2 vectors

The unit vector isobtained by dividing by the modulus.

The modulus is $∥〈0,-16,-12〉∥=sqrt(0+16^2+12^2)=sqrt400=20$

The unit vector is $=1/20〈0,-16,-12〉=1/5〈0,-4,-3〉$

How do you find a unit vector that is orthogonal to a and b where a = −7 i + 6 j − 8 ka=−7i+6j−8ka = −7 i + 6 j − 8 k and b = −5 i + 6 j − 8 kb=−5i+6j−8kb = −5 i + 6 j − 8 k?