Question

Find angle between lines who Dr's are given by $(1,1,2)(\sqrt{31}-\sqrt{310})$.

Answer 1

Angle between the lines, when Dr's are given is:

$\cos \theta =\left|\frac{a_1{a}_2+b_1{b}_2+c_1{c}_2}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}\right|$

From given equations:

$a_1=1,b_1=1,c_1=2$ and $a_2=\sqrt{31}-\sqrt{310},b_2=0,c_2=0$

$\cos \theta =\left|\frac{1\times \sqrt{31}-\sqrt{310}+1\times 0+1\times 0}{\sqrt{1^2+1^2+2^2}\sqrt{(\sqrt{31}-\sqrt{310}{)}^2+0^2+0^2}}\right|$

$\cos \theta =\left|\frac{\sqrt{31}-\sqrt{310}}{\sqrt{6}(\sqrt{31}-\sqrt{310})}\right|$

$\cos \theta =\frac{1}{\sqrt{6}}$

$\theta =65.90$