How do you find a unit vector orthogonal to both $(2,0,1,-4)$ and $(2,3,0,1)$ ?

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Answer 1 · 2016-07-12T15:44:29

See below

Calling $vec u = {2,0,1,-4}$ and $vec v = {2,3,0,1}$

we need a vector

$vec x = {a,b,c,d}$

such that

$<< vec u, vec x >> = 0$
$<< vec v, vec x >> = 0$
$norm (vec x) = 1$

Solving

${ (2 a + c - 4 d = 0), (2 a + 3 b + d = 0), (a^2 + b^2 + c^2 + d^2 = 1) :}$

for $a,b,c$ we obtain

$((a = 1/7 (10 d - 3 sqrt[1 - 6 d^2])),( b = 1/7 (-9 d + 2 sqrt[1 - 6 d^2])), (c = 2/7 (4 d + 3 sqrt[1 - 6 d^2])))$

or

$((a = 1/7 (10 d + 3 sqrt[1 - 6 d^2])),( b = 1/7 (-9 d - 2 sqrt[1 - 6 d^2])), (c = 2/7 (4 d - 3 sqrt[1 - 6 d^2])))$

then if we choose $1-6d^2 ge 0$ or $-1/sqrt(6) le d le 1/sqrt(6)$ we will have solutions to this problem

How do you find a unit vector orthogonal to both (2,0,1,-4)(2,0,1,-4) and (2,3,0,1)(2,3,0,1)?