Question

If the direction cosines of a line are \(\dfrac{4}{\sqrt{77}},\ \dfrac{5}{\sqrt{77}},\ \dfrac{x}{\sqrt{77}}\), then the value of \(x\) is

• A. \(6\)
• B. \(7\)
• C. \(8\)
• D. \(9\)

Hint:

Direction cosines are the components of a unit direction vector, so they always satisfy \(l^2+m^2+n^2=1\).

Answer 1

The Correct Option is A

For any line in 3-D, its direction cosines \(l,m,n\) satisfy \[ l^{2}+m^{2}+n^{2}=1 \text{(unit vector condition).} \] Here \(l=\dfrac{4}{\sqrt{77}},\ m=\dfrac{5}{\sqrt{77}},\ n=\dfrac{x}{\sqrt{77}}\). So \[ \left(\frac{4}{\sqrt{77}}\right)^{2}+\left(\frac{5}{\sqrt{77}}\right)^{2}+\left(\frac{x}{\sqrt{77}}\right)^{2}=1 \] \[ \frac{16+25+x^{2}}{77}=1 \ \Rightarrow\ 16+25+x^{2}=77\ \Rightarrow\ x^{2}=36. \] Hence \(x=\pm 6\). Among the choices the matching value is \(6\).