Question

The direction cosines \(l,m,n\) of the two lines are connected by the relations \(l+m+n=0\) and \(lm=0\), then the angle between the lines is....

• A. \(0\)
• B. \(45^{\circ }\)
• C. \(60^{\circ }\)
• D. \(90^{\circ }\)

Hint:

From lm = 0 take l = 0 or m = 0, find each line's direction ratios, then use the cosine of the angle.

Answer 1

The Correct Option is C

Step 1: Equal Magnitudes
Both lines have direction ratios of magnitude $\sqrt{2}$, so the direction cosines are $\left(0,\tfrac{1}{\sqrt{2}},-\tfrac{1}{\sqrt{2}}\right)$ and $\left(\tfrac{1}{\sqrt{2}},0,-\tfrac{1}{\sqrt{2}}\right)$.

Step 2: Dot Product
$\cos\theta=0+0+\dfrac{1}{2}=\dfrac{1}{2}$.

Step 3: Angle
$\theta=60^{\circ}$.

Final Answer:
The lines meet at 60 degrees. \[ \boxed{\text{(C) }60^{\circ}} \]