MHT CET · Maths · Three Dimensional Geometry
The angle between a line with direction ratios 2, 2, 1 and a line joining \((3,1,4)\) and \((7,2,12)\) is
- A \(\cos ^{-1}\left(\frac{1}{\sqrt{3}}\right)\)
- B \(\cos ^{-1}\left(\frac{1}{3}\right)\)
- C \(\cos ^{-1}\left(\frac{2}{3}\right)\)
- D \(\cos ^{-1}\left(\frac{\sqrt{2}}{3}\right)\)
Answer & Solution
Correct Answer
(C) \(\cos ^{-1}\left(\frac{2}{3}\right)\)
Step-by-step Solution
Detailed explanation
Direction ratios of a line joining \((3,1,4)\) and \((7,2,12)\) are \(4,1,8\)
Let \(\left(a_1, b_1, c_1\right)=(2,2,1)\) and \(\left(a_2, c_2\right)=(4,1,8)\).
Hence angle \(\theta\) between the lines is given by
\(\cos \theta=\frac{a_1 a_2+b_1 b_2+c_1 c_2}{\sqrt{a_1^2+b_1^2+c_1 \sqrt{a_2^2+b_2^2+c_2^2}}}\)
\(=\frac{8+2+8}{\sqrt{4+4+1} \cdot \sqrt{16+1+64}}=\frac{18}{\sqrt{9} \cdot \sqrt{81}}=\frac{18}{(3)(9)}=\frac{2}{3}\)
\(\therefore \theta=\cos ^{-1}\left(\frac{2}{3}\right)\)
Let \(\left(a_1, b_1, c_1\right)=(2,2,1)\) and \(\left(a_2, c_2\right)=(4,1,8)\).
Hence angle \(\theta\) between the lines is given by
\(\cos \theta=\frac{a_1 a_2+b_1 b_2+c_1 c_2}{\sqrt{a_1^2+b_1^2+c_1 \sqrt{a_2^2+b_2^2+c_2^2}}}\)
\(=\frac{8+2+8}{\sqrt{4+4+1} \cdot \sqrt{16+1+64}}=\frac{18}{\sqrt{9} \cdot \sqrt{81}}=\frac{18}{(3)(9)}=\frac{2}{3}\)
\(\therefore \theta=\cos ^{-1}\left(\frac{2}{3}\right)\)
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
- If \(\theta\) and \(\alpha\) are not odd multiples of \(\frac{\pi}{2}\) then \(\tan \theta=\tan \alpha\) implies principal solution isMHT CET 2024 Easy
- With usual notations, in \(\triangle A B C\), the lengths of two sides are 10 cm and 9 cm respectively. If angles \(A, B, C\) are in A.P. then perimeter of ABC isMHT CET 2025 Medium
- If \(x^{2}-2 p x y-y^{2}=0\) and \(x^{2}-2 q x y-y^{2}=0\)
bisect angles between each other, thenMHT CET 2007 Medium - If \(y=\cos ^{-1}\left(\frac{\mathrm{a}^2}{\sqrt{x^4+\mathrm{a}^4}}\right)\), then \(\frac{\mathrm{d} y}{\mathrm{~d} x}\) isMHT CET 2023 Medium
- The general solution of the differential equation. \(\frac{\mathrm{dy}}{\mathrm{d} x}+\sin \left(\frac{x+\mathrm{y}}{2}\right)=\sin \left(\frac{x-\mathrm{y}}{2}\right)\) isMHT CET 2025 Medium
- If \(a, b, c\) are distinct positive numbers and vectors \(a \hat{\imath}+a \hat{\jmath}+c \hat{k}, \hat{\imath}+\hat{k}\)
and \(c \hat{\imath}+c \hat{\jmath}+b \hat{k}\) lie in a plane, thenMHT CET 2020 Easy
More PYQs from MHT CET
- Let \(2 \sin ^2 x+3 \sin x-2\gt0\) and \(x^2-x-2 \lt 0\). ( \(x\) is measured in radians). The \(x\) lies in the intervalMHT CET 2024 Hard
- If the points \(\mathrm{P}(4,5, \mathrm{x}), \mathrm{Q}(3, \mathrm{y}, 4)\) and \(\mathrm{R}(5,8,0)\) are collinear, then the value of \(x+y\) isMHT CET 2021 Medium
- The correct order of dehydration of alcohols isMHT CET 2010 Easy
- In \(\mathrm{P}^{\text {th }}\) second, a particle describes angular displacement of ' \(\beta\) ' rad. If it starts from rest, the angular acceleration isMHT CET 2023 Hard
- If then _______MHT CET 2016 Easy
- The blood vessel indicated by 'X' is
MHT CET 2019 Medium