MHT CET · Maths · Three Dimensional Geometry
The acute angle between the lines \(x=-y, z=0\) and \(x=0, z=0\) is
- A \(\frac{\pi}{3}\)
- B \(\frac{\pi}{6}\)
- C \(\frac{\pi}{4}\)
- D \(\frac{\pi}{18}\)
Answer & Solution
Correct Answer
(C) \(\frac{\pi}{4}\)
Step-by-step Solution
Detailed explanation
Given lines are
\(\frac{x-0}{1}=\frac{y-0}{-1}=\frac{z-0}{0} \text { and } \frac{x-0}{0}=\frac{y-0}{1}=\frac{z-0}{0}\)
\(\text { Now, the angle }=\cos ^{-1}\left(\frac{a_1 a_2+b_1 b_2+c_1 c_2}{\sqrt{a_1^2+b_1^2+c_1^2} \cdot \sqrt{a_2^2+b_2^2+c_2^2}}\right)\)
\(=\cos ^{-1}\left(\frac{1 \times 0+(-1) \times 1+0 \times 0}{\sqrt{1^2+(-1)^2+0^2} \cdot \sqrt{0^2+1^2+0^2}}\right)\)
\( =\cos ^{-1}\left|\frac{-1}{\sqrt{2}}\right|[\text { for acute angle }]\)
\(=\cos ^{-1}\left(\frac{1}{\sqrt{2}}\right)=\frac{\pi}{4}\)
\(\frac{x-0}{1}=\frac{y-0}{-1}=\frac{z-0}{0} \text { and } \frac{x-0}{0}=\frac{y-0}{1}=\frac{z-0}{0}\)
\(\text { Now, the angle }=\cos ^{-1}\left(\frac{a_1 a_2+b_1 b_2+c_1 c_2}{\sqrt{a_1^2+b_1^2+c_1^2} \cdot \sqrt{a_2^2+b_2^2+c_2^2}}\right)\)
\(=\cos ^{-1}\left(\frac{1 \times 0+(-1) \times 1+0 \times 0}{\sqrt{1^2+(-1)^2+0^2} \cdot \sqrt{0^2+1^2+0^2}}\right)\)
\( =\cos ^{-1}\left|\frac{-1}{\sqrt{2}}\right|[\text { for acute angle }]\)
\(=\cos ^{-1}\left(\frac{1}{\sqrt{2}}\right)=\frac{\pi}{4}\)
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Maths
- Rajesh has just bought a VCR from Maharashtra Electronics and the shop offers after sales service contract for Rs. 1000 for the next five years. Considering the experience of VCR users, the following distribution of maintenance expenses for the next five years is formed.
The expected value of maintenance cost isMHT CET 2021 Easy - If slope of the tangent to the curve \(x y+a x+b y=0\) at the point \((1,1)\) on it is 2 , then the value of \(3 a+b\) isMHT CET 2023 Easy
- The value of \(\tan ^{-1}(-\sqrt{3})-\sin ^{-1}\left(\frac{1}{\sqrt{2}}\right)+\cos ^{-1}\left(\frac{-1}{2}\right)\) isMHT CET 2024 Easy
- The shortest distance between the lines
\(\begin{aligned} & \overline{\mathrm{r}}=(4 \hat{i}-\hat{j})+\lambda(\hat{i}+2 \hat{j}-3 \hat{k}) \text { and } \\ & \overline{\mathrm{r}}=(\hat{i}-\hat{j}+2 \hat{k})+\mu(2 \hat{i}+4 \hat{j}-5 \hat{k}) \text { is }\end{aligned}\)MHT CET 2025 Medium - Let \(O\) be the origin and let \(P Q R\) be an arbitrary triangle. The point \(S\) \(\overline{O P} \cdot \overline{O Q}+\overline{O R} \cdot \overline{O S}=\overline{O R} \cdot \overline{O P}+\overline{O Q} \cdot \overline{O S}=\) \(\overline{O Q} \overline{O Q} \cdot \overline{O R}+\overline{O P} \cdot \overline{O S}\) that \(\overline{O P} \cdot \overline{O Q}+\overline{O R} \cdot \overline{O S}=\overline{O R} \cdot \overline{O P}+\overline{O Q} \cdot \overline{O S}=\) \(\overline{O Q} \cdot \overline{O R}+\overline{O P} \cdot \overline{O S}\), then the triangle \(P Q R\) has \(S\) as itsMHT CET 2022 Medium
- The vector projection of \(\overline{\mathrm{AB}}\) on \(\overline{\mathrm{CD}}\), where \(A \equiv(2,-3,0), B \equiv(1,-4,-2), C \equiv(4,6,8)\) and \(\mathrm{D} \equiv(7,0,10)\), isMHT CET 2023 Easy
More PYQs from MHT CET
- The magnetic field inside a current carrying toroidal solenoid is \(0.2 \mathrm{mT}\). What is the magnetic field inside the toroid if the currentMHT CET 2021 Medium
- Which among the following is dicarboxylic acid?MHT CET 2025 Easy
- Complete the analogy ANP : inhibition of JGA cells :: Angiotensin II : ______MHT CET 2024 Hard
- Two radioactive materials ' \(M_1{ }^{\prime}\) and ' \(M_2{ }^{\prime}\) have decay constants ' \(9 \lambda\) ' and ' \(\lambda\) ' respectively. Initially they have the same number of nuclei. The ratio of the number of nuclei of \(M_1\) to that of \(M_2\) will be ' \(\left(\frac{1}{e}\right)^{\prime}\) after a time intervalMHT CET 2025 Medium
- Which from following compounds is obtained when phenol reacts with dilute nitric acid at low temperature?MHT CET 2025 Easy
- When a photon enters glass from air, which one of the following quantities does not change?MHT CET 2021 Easy