English
Hindi
Bengali
Marathi
Tamil

Question

Download Solution PDF

Find the angle between the line  and the plane 2x - 3y + z - 5 = 0 ?

Answer (Detailed Solution Below)

Option 4 :
Free Tests
View all Free tests >
Free
Agniveer Navy SSR Full Test - 01
100 Qs. 100 Marks 60 Mins
Start Now

Detailed Solution

Download Solution PDF

Concept:

If θ is the angle between the line  and the plane a2x + b2 y + c2z + d = 0 then

Calculation:

Given: Equation of line is  and equation of plane is 2x - 3y + z - 5 = 0.

As we know that the angle between the line  and the plane a2x + b2 y + c2z + d = 0  is given by: 

Here, a1 = 1, b1 = - 2, c1 = - 3, a2 = 2, b2 = - 3 and c2 = 1.

⇒ a1⋅ a2 + b1 ⋅ b2 + c1 ⋅ c2 = 2 + 6 - 3 = 5

 and 

Download Solution PDF
Latest Indian Navy Agniveer SSR Updates

Last updated on Jun 20, 2025

-> The Indian Navy SSR Agniveeer Merit List has been released on the official website.

-> The Indian Navy Agniveer SSR CBT Exam was conducted from 25th to 26th May 2025.

->The application window was open from 29th March 2025 to 16th April 2025.  

-> The Indian Navy SSR Agniveer Application Link is active on the official website of the Indian Navy.

.->Only unmarried males and females can apply for Indian Navy SSR Recruitment 2025.

-> The selection process includes a Computer Based Test, Written Exam, Physical Fitness Test (PFT), and  Medical Examination.

->Candidates Qualified in 10+2 with Mathematics & Physics from a recognized Board with a minimum 50% marks in aggregate can apply for the vacancy.

-> With a salary of Rs. 30,000, it is a golden opportunity for defence job seekers.

-> Candidates must go through the Indian Navy SSR Agniveer Previous Year Papers and Agniveer Navy SSR mock tests to practice a variety of questions for the exam.

Exams Tests Super Practice

More Angle with Planes Questions

Q1.If the angle θ between the line \(\rm \frac{x+1}{1}=\frac{y-1}{2}=\frac{z-2}{2}\) and plane 2x - y + √λ z + 4 = 0 is such that sin θ = \(\frac{1}{3}\), then value of λ is:
Q2.The plane ax +  by = 0 is rotated through an angle α. The equation of plane in its new position is:
Q3.The plane 2x – 3y + 6z – 11 = 0 makes an angle sin–1(α) with x-axis. The value of α is equal to
Q4.The locus represented by xy + yz = 0 is
Q5.The sine of the angle between the straight line \(\rm \frac{x-2}{3}=\frac{y-3}{4}=\frac{z-4}{5}\) and the plane 2x − 2y + z = 5 is
Q6.Find a vector perpendicular to the plane that passes through the points P(1, 4, 6), Q(-2, 5, -1) and R(1, -1, 1).
Q7.Find the angle between the planes x + y + z = 1 and x - 2y + 3z = 1.
Q8.The minute hand of a watch is 2 cm long. How far does its tip move in 45 minutes? (use π = 3.14) 
Q9.Determine the angle between the two planes 2x + y - 2z = 3 and 3x – 6y – 2z = 9 using vector method?
Q10.March List I with List II :   List I   List II (A) Supplementary angles (I) Angles less than 90° (B) Complementary angles (II) Two angles whose sum is 180° (C) Acute angles (III) Angles greater than 90° but less than 180° (D) Obtuse angles (IV) Two angles whose sum is 90° Choose the correct answer from the options given below :

More Three Dimensional Geometry Questions

Q1.The distance of the point (1, 1, 9) from the point of intersection of the line (x – 3)/1 = (y – 4)/2 = (z – 5)/2 and the plane x + y + z = 17 is:
Q2.The equation of the plane passing through the point (1, 2, –3) and perpendicular to the planes 3x + y – 2z = 5 and 2x – 5y – z = 7, is:
Q3.Let P be a plane passing through the points (2, 1, 0), (4, 1, 1) and (5, 0, 1) and R be any point (2, 1, 6). Then the image of R in the plane P is:
Q4.Let a, b∈R. If the mirror image of the point P(a, 6, 9) with respect to the line (x – 3)/7 = (y – 2)/5 = (z – 1)/(-9) is (20, b, -a -9), then |a + b| is equal to:
Q5.Let P be the plane containing the straight line \(\frac{x-3}{9}=\frac{y+4}{-1}=\frac{z-7}{-5}\) and perpendicular to the plane containing the straight lines \(\frac{x}{2}=\frac{y}{3}=\frac{z}{5}\) and \(\frac{x}{3}=\frac{y}{7}=\frac{z}{8}\) If d is the distance P from the point (2, –5, 11), then d2 is equal to :
Q6.A square is inscribed in the circle x2 + y2 – 10x – 6y + 30 = 0. One side of this square is parallel to y = x + 3. If (xi, yi) are the vertices of the square, then \(\sum(x_i^2 + y_i^2)\) is equal to:
Q7.If the eqation of the plane containing the line x + 2y + 3z – 4 = 0 = 2x + y –z + 5 and perpendicular to the plane\(\vec{r}=(\hat{i}-\hat{j})+\lambda(\hat{i}+\hat{j}+\hat{k})+\mu(\hat{i}-2 \hat{j}+3 \hat{k})\) is ax + by + cz = 4, then (a – b + c) is equal to
Q8.If the points with position vectors αî + 10ĵ + 13k̂, 6î +11ĵ +11k̂, \( \frac{9}{2} \)î + βĵ - 8k̂  are collinear, then (19α – 6β)2 is equal to 
Q9.The shortest distance between the lines \(\frac{x-4}{4}=\frac{y+2}{5}=\frac{z+3}{3} \) and \( \frac{x-1}{3}=\frac{y-3}{4}=\frac{z-4}{2} \) is
Q10.Let C(α, β) be the circumcenter of the triangle formed by the lines 4x + 3y = 69, 4y – 3x = 17 and x + 7y = 61.  Then (α – β)2 + α + β is equal to 
Suggested Test Series
View All >
General Knowledge for All Defence Exams
317 Total Tests with 4 Free Tests
Start Free Test
NCERT XI-XII Physics Foundation Pack Mock Test
323 Total Tests with   5 Free Tests
Start Free Test

Indian Navy Agniveer SSR Important Links

Admit CardEligibility CriteriaResultSyllabus and Exam PatternCut OffBooks

More Mathematics Questions

Q1.log10 10000 = _______?
Q2.Consider a system equation \(2x+y-z=0\), \(4x - py + 4z = 4\) and \(x-y+z=q\) where \(p, q \in I\) and \(p, q \in [1, 10]\), then identify the correct statement(s).   List-I List-II (I) Number of ordered pairs \((p, q)\) for which system of equation has unique solution is (P) 1 (II) Number of ordered pairs \((p, q)\) for which system of equation has no solution is (Q) 9 (III) Number of ordered pairs \((p, q)\) for which system of equation has infinite solution is (R) 91 (IV) Number of ordered pairs \((p, q)\) for which system of equation has atleast one solution is (S) 90
Q3.Match List I with List II:   List - I List - II (I) The number of solutions of the equation\((1 - \cos 2x)(\cos 2x + \cot^2 x) = 0\) where \(0 \le x \le 2\pi\) is (P) 4 (II) \(\tan 81^\circ - \tan 63^\circ - \tan 27^\circ + \tan 9^\circ\) equals (Q) 3 (III) The number of solutions of (x, y), which satisfy |y| = sin x and y = cos-1(cos x), where -2π ≤ x ≤ 2π is (R) 2 (IV) For the equation \(\cos^{-1}x + \cos^{-1}2x + \pi = 0\), the number of real solutions is (S) 1     (T) 0  
Q4.If Σn = 36, find n
Q5.If log 2 = 0.2614, log 3 = 0.3521, log 6 = ?
Q6.The growth model for a population is Nt = N0ert where Nt is the population at time t, N0 is the initial population and r is the per capita growth rate. How long does it take for the population numbers to double?
Q7.Suppose that the growth rate of the number of individuals (N) in a population in an environment with a carrying capacity K is modelled as \(\frac{d N}{d t}=r N \frac{K-N}{K}\) where r is the per capita growth rate and t is time; r and K are positive constants. At which of the following values of N will the population growth rate be the maximum?
Q8.The shadow of a vertical pole, formed due to a nearby lamp which is at a height of 12 m, measures 4 m. If the lamp is moved horizontally 2 m further away from the pole, the length of the shadow measures 8 m. What is the height (in m) of the pole?
Q9.A pack of cards contains 4 aces, 4 kings, 4 queens and 4 jacks. Out of these 16 number of cards, two cards are drawn at random. Find the probability that at least one of them is an ace :
Q10.If the equations 2x2 + kx - 5 = 0 and x2 - 3x - 4 = 0 have one root common, then find the value of k.
Testbook Edu Solutions Pvt. Ltd.
1st & 2nd Floor, Zion Building,
Plot No. 273, Sector 10, Kharghar,
Navi Mumbai - 410210
support@testbook.com
Toll Free:1800 833 0800
Office Hours: 10 AM to 7 PM (all 7 days)
Company
About usCareers We are hiringTeach Online on TestbookPartnersMediaSitemap
Products
Test SeriesTestbook PassOnline CoursesOnline VideosPracticeBlogRefer & EarnBooks
Copyright © 2014-2022 Testbook Edu Solutions Pvt. Ltd.: All rights reserved
User PolicyTermsPrivacy
Hot Links： teen patti lotus teen patti lucky teen patti game paisa wala teen patti neta