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# Answer on Linear Algebra Question for biju

Question #2167
If u = 2i &minus; 3j + k, v = i &minus; 2k, w = ai + bj + ck form an orthogonal basis of R3 , find the
possible values of a, b and c.
Further, obtain the angles x = i &minus; 3j&minus; 3k makes with each of these vectors.
b) Find the direction cosines of the perpendicular from the origin to the plane
3r.(2i &minus; 3j+ k) + 7 = 0.
Let&#039;s construct the system of equations for a,b,c. By the definition of the orthonormal basis:

$\vec{u}\cdot \vec{w} = \vec{v}\cdot\vec{w}\equiv 0\\ |\vec{w}|=1.\\ \vec{u} =[2,-3,1], \ \vec{v}=[1,0,-2], \ \vec{w}=[a,b,c] \\ \\ \vec{u}\cdot \vec{w} = 2a -3b + c =0 \\ \vec{v}\cdot \vec{w} = a - 2c =0 \\ a^2+b^2+c^2 = 1\\ \\ a = 2c, \ b = \frac{5}{3}c, \\ c^2(4 + \frac{25}{9} +1) = 1 \\ c = 3/\sqrt{70}, \ a = 6/\sqrt{70}, \ b = 5/\sqrt{70}$

The angles can be obtained in the following way:
$|\vec{x}| = \sqrt{1+9+9}= \sqrt{19} \hspace{6mm} |\vec{u}|=|\vec{v}|=|\vec{w}|=1 \\ angle[x u] = \arccos{(\vec{x}\cdot\vec{u})/|x|} = \arccos {\frac{8}{\sqrt{19}}}\\ angle[x v] = \arccos{(\vec{x}\cdot\vec{v})/|x|} = \arccos {-\frac{5}{\sqrt{19}}} \\ angle[x w] = \arccos{(\vec{x}\cdot\vec{w})/|x|} = \arccos {-\frac{18}{\sqrt{19}\sqrt{70}}} \\$
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