Question

The angle between the lines joining the points $(1,1,0),(-3,\sqrt{3}+1,3)\text{and}(0,-1,0),(-1,\sqrt{3}-1,\lambda )$ is ${\cos }^{-1}\left(\sqrt{\frac{7}{16}}\right)$.If $\lambda$ is an integer then $\lambda$ is:

Answer 1

Let points are $A(1,1,0),B(-3,\sqrt{3}+1,3)\text{and}C(0,-1,0),D(-1,\sqrt{3}-1,\lambda )$
D.R's of $\overrightarrow{AB}=(-4,\sqrt{3},3)$;
D.R's of $\overrightarrow{CD}=(-1,\sqrt{3},\lambda )\cos \theta =\frac{\overline{AB}.\overline{CD}}{|\overline{AB}||\overline{CD}|}\sqrt{\frac{7}{16}}=\frac{4+3+3\lambda }{\sqrt{16+3+9}\sqrt{1+3+{\lambda }^2}}$
On simplifying and solving, we have:
$13{\lambda }^2-168\lambda =0\Rightarrow \lambda =0,\lambda =\frac{168}{13}$
Hence, the integer value of $\lambda$ is $0$.