Lines L₁, L₂, L₃ have direction ratios 2, -1, 2; 2, 2, -1; and -1, 2, 2, respectively. Which of the following statements are true? I. L₁ is perpendicular to L₂. II. L₁, L₂ and L₃ are mutually perpendicular. III. L₃ is parallel to the line with direction ratios 1, -2, -2. IV. L₁ is parallel to the line with direction ratios 4, -2, -4.

Question 76

MediumOriginal question

Lines $L_1$ , $L_2$ , $L_3$ have direction ratios $2, -1, 2$ ; $2, 2, -1$ ; and $-1, 2, 2$ , respectively. Which of the following statements are true?
I. $L_1$ is perpendicular to $L_2$ .
II. $L_1$ , $L_2$ and $L_3$ are mutually perpendicular.
III. $L_3$ is parallel to the line with direction ratios $1, -2, -2$ .
IV. $L_1$ is parallel to the line with direction ratios $4, -2, -4$ .

I, II and III onlyCorrect answerYour answerAll four: I, II, III and IVCorrect answerYour answerI and II onlyCorrect answerYour answerI and III onlyCorrect answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer A

Correct answer: A

For $L_1, L_2$ : $4 - 2 - 2 = 0$ ; for $L_1, L_3$ : $-2 - 2 + 4 = 0$ ; for $L_2, L_3$ : $-2 + 4 - 2 = 0$ . So I and II are true.
$-1, 2, 2 = -(1, -2, -2)$ , so III is true; twice $2, -1, 2$ is $4, -2, 4$ , not $4, -2, -4$ , so IV is false.

Worth noting

IV fails on one sign: twice $2, -1, 2$ is $4, -2, 4$ .

Concept: Parallel and perpendicular lines
Typical timeTypical: 120 s
Your timeYours: —

The rule this testsThe angle between two lines

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