Question

If the angle between two intersecting lines having direction ratios $(5,7,3)$ & $(3,4,5)$ respectively can be given by
${\cos }^{-1}\left(\frac{58}{\sqrt{b}}\right)$,
then what will be the value of $b$ ?

• A. 4150
• B. 4052
• C. 3971
• D. 4167

Answer 1

The correct option is A 4150

We know that If two lines with direction cosines $l_1,m_1,n_1$ and $l_2,m_2,n_2$ intersect then the angle between them would be ${\cos }^{-1}(l_1.l_2+m_1.m_2+n_1.n_2)$ Here we are given direction ratios and not the direction cosines. So we’ll find direction cosines first.
Direction cosines for the first line will be -
$\frac{5}{\sqrt{5^2+7^2+3^2}},\frac{7}{\sqrt{5^2+7^2+3^2}},\frac{3}{\sqrt{5^2+7^2+3^2}}$
Or, $\frac{5}{\sqrt{83}},\frac{7}{\sqrt{83}},\frac{3}{\sqrt{83}}$
Similarly, direction cosines for the second line will be -
$\frac{3}{\sqrt{3^2+4^2+5^2}},\frac{4}{\sqrt{3^2+4^2+5^2}},\frac{5}{\sqrt{3^2+4^2+5^2}}$
Or, $\frac{3}{\sqrt{50}},\frac{4}{\sqrt{50}},\frac{5}{\sqrt{50}}$
Now we can calculate the angle between these lines, which will be -
${\cos }^{-1}(\frac{5}{\sqrt{83}}.\frac{3}{\sqrt{50}}+\frac{7}{\sqrt{83}}.\frac{4}{\sqrt{50}}+\frac{3}{\sqrt{83}}.\frac{5}{\sqrt{50}})$
Or, ${\cos }^{-1}\left(\frac{58}{\sqrt{4150}}\right)$
On comparing it with the given expression we get b = 4150