Consider two lines $l_1$ and $l_2$ with cartesian equations $\frac{x}{2}=\frac{1-y}{-2}=\frac{z}{1}$ and $\frac{2x-5}{16}=\frac{y-2}{-1}=\frac{x-5}{4}$ respectively. Which of the following is/are true?
(A) Direction ratio of $l_1$ are 2, 2, 1
(B) Direction cosines of $l_1$ are $\frac{2}{3},\frac{-2}{3},\frac{1}{3}$
(C) Direction ratio of $l_2$ are 16, -1, 4
(D) Angle between $l_1$ and $l_2$ is $\cos^{-1}(\frac{38}{3\sqrt{273}})$
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A) only
Given
$l_{1}:\frac{x}{2}=\frac{1-y}{-2}=\frac{z}{1}$
Rewrite:
$\frac{x-0}{2}=\frac{y-1}{2}=\frac{z-0}{1}$
So direction ratios of $l_{1}$ are $(2,2,1)$.
(A) is true.
Direction cosines of $l_{1}$:
Magnitude $=\sqrt{2^{2}+2^{2}+1^{2}}=3$
So DCs $=\left(\frac{2}{3},\frac{2}{3},\frac{1}{3}\right)$
But option (B) gives $\left(\frac{2}{3},-\frac{2}{3},\frac{1}{3}\right)$, so (B) is false.
Given
$l_{2}:\frac{2x-5}{16}=\frac{y-2}{-1}=\frac{x-5}{4}$
Rewrite:
$\frac{x-\frac{5}{2}}{8}=\frac{y-2}{-1}=\frac{x-5}{4}$
This is inconsistent in variables; as DRs must be a multiple of DC's.
(C) is false.
Angle between $l_{1}$ and $l_{2}$:
$\cos\theta=\frac{2\cdot16+2(-1)+1\cdot4}{\sqrt{(2^{2}+2^{2}+1^{2})(16^{2}+(-1)^{2}+4^{2})}}$
$=\frac{32-2+4}{\sqrt{9(256+1+16)}}$
$=\frac{34}{3\sqrt{273}}$
So (D) is false.
Final answer: (A) only