Just take any 3 independent vectors $vec v_1,vec v_2,vec v_3$, additionally independent of $vec v$. You might usage the conventional basis vectors (<1,0,0,0> etc.). Then project them away from $vec v$.
You are watching: Find a basis of the subspace of r4 that consists of all vectors perpendicular to both
$$vec b_1 = vec v_1 - left(fracvec v_1cdotvec vvec vcdotvec v ight)vec v$$$$vec b_2 = vec v_2 - left(fracvec v_2cdotvec vvec vcdotvec v ight)vec v$$$$vec b_3 = vec v_3 - left(fracvec v_3cdotvec vvec vcdotvec v ight)vec v$$
You can use the Gram-Schmidt procedure rather, if you desire an orthonormal basis.
I think I have figured it out. When you take v and among it"s orthogonal vectors the dot product will certainly provide you zero, therefore let the orthogonal vector = (a,b,c,d) and let v = (e,f,g,h), then ae+bf+cg+dh = 0 then if you isolate a you have the right to get a = ((-f)b+(-g)c+(-h)d)/e, you can then relocation a in your orthogonal vector giving
((-f)b+(-g)c+(-h)d)/e , b , c , d)/ which can the be decomposed into
b ((-f)/e , 1 , 0 , 0) + c ((-g)/e , 0 , 1 , 0) + d ((-h)/e , 0 , 0 , 1)
P.S. Sorry for the lack of notation and structure I"m not quite provided to proofs yet.
You want to find three livirtually independent vectors, which are perpendicular to $v$
For instance you might take into consideration
in situation of $v_i e 0$ for $i=1,2,3,4$
Thanks for contributing an answer to thedesigningfairy.comematics Stack Exchange!Please be sure to answer the question. Provide details and share your research!
But avoid …Asking for assist, clarification, or responding to other answers.Making statements based on opinion; ago them up through references or individual endure.
Use thedesigningfairy.comJax to format equations. thedesigningfairy.comJax recommendation.
See more: During Middle Age, Fibrous Tissue Within Sutures Ossifies, Leaving Closed Sutures Called
To learn even more, view our tips on writing excellent answers.
Article Your Answer Discard
Not the answer you're looking for? Browse various other inquiries tagged linear-algebra or ask your very own question.
What does "the orthogonal basis vectors covering the subarea perpendicular to vector $vece_1$" mean?
website style / logo design © 2021 Stack Exchange Inc; user contributions licensed under cc by-sa. rev2021.9.14.40215