Question

Given below are two statements:

Assertion (A):
The angle between the pair of lines \[ \frac{x+3}{3}=\frac{y-1}{5}=\frac{z+3}{4} \quad \text{and} \quad \frac{x+1}{1}=\frac{y-4}{1}=\frac{z-5}{2} \] is \(\cos^{-1}\left(\frac{8\sqrt{3}}{15}\right)\).
Reason (R):
The angle between the two lines is \[ \cos\theta=\frac{a_1a_2+b_1b_2+c_1c_2}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}. \]

• A. Both (A) and (R) are correct and (R) is the correct explanation of (A)
• B. Both (A) and (R) are correct but (R) is not the correct explanation of (A)
• C. (A) is correct but (R) is not correct
• D. (A) is not correct but (R) is correct

Hint:

For angle between two lines in 3D, use direction ratios and apply the dot product formula.

Answer 1

The Correct Option is A

Concept:
The angle between two lines in three-dimensional geometry is found using their direction ratios.

Step 1: Identify the direction ratios of first line. \[ \frac{x+3}{3}=\frac{y-1}{5}=\frac{z+3}{4} \] So, direction ratios of first line are: \[ a_1=3,\quad b_1=5,\quad c_1=4 \]

Step 2: Identify the direction ratios of second line. \[ \frac{x+1}{1}=\frac{y-4}{1}=\frac{z-5}{2} \] So, direction ratios of second line are: \[ a_2=1,\quad b_2=1,\quad c_2=2 \]

Step 3: Use the formula for angle between two lines. \[ \cos\theta=\frac{a_1a_2+b_1b_2+c_1c_2}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}} \] \[ \cos\theta=\frac{3(1)+5(1)+4(2)}{\sqrt{3^2+5^2+4^2}\sqrt{1^2+1^2+2^2}} \] \[ \cos\theta=\frac{3+5+8}{\sqrt{9+25+16}\sqrt{1+1+4}} \] \[ \cos\theta=\frac{16}{\sqrt{50}\sqrt{6}} \] \[ \cos\theta=\frac{16}{\sqrt{300}} \] \[ \cos\theta=\frac{16}{10\sqrt{3}} \] \[ \cos\theta=\frac{8}{5\sqrt{3}} \] Rationalizing, \[ \cos\theta=\frac{8\sqrt{3}}{15} \] \[ \theta=\cos^{-1}\left(\frac{8\sqrt{3}}{15}\right) \] Therefore, Assertion is correct. Reason gives the correct formula and also explains Assertion. \[ \therefore \text{Correct Answer is (A)} \]