Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
JEE Main202330 Jan 2023Evening ShiftMathematicsThree Dimensional GeometryActual

A vector v → in the first octant is inclined to the x axis at 60 ° , to the y -axis at 45 ° and to the z -axis at an acute angle. If a plane passing through the points 2 , - 1 , 1 and a , b , c , is normal to v → , then

Options

  1. A2 a + b + c = 1
  2. Ba + b + 2 c = 1
  3. Ca + 2 b + c = 1
  4. D2 a - b + c = 1

Correct answer

C. a + 2 b + c = 1

Step-by-step solution

Direction cosines l , m   a n d   n of given vector v → , l = cos 60 ∘ = 1 2 m = cos 45 ∘ = 1 2 n = cos γ v → = cos 60 ° i ^ + cos 45 ° j ^ + cos γ k ^ . . . . ( 1 ) We know, l 2 + m 2 + n 2 = 1 ⇒ 1 4 + 1 2 + cos 2 γ = 1    ( γ → Acute ) ⇒ cos γ = 1 2 ⇒   γ = 60 ° Equation of plane is 1 2 x - 2 + 1 2 y + 1 + 1 2 z - 1 = 0 ⇒ x 2 + y 2 + z 2 = 1 2 ⇒     x + 2 y + z = 1 a , b , c

Practice Three Dimensional Geometry on Quantrex Academy →

More from Three Dimensional Geometry

Let P be the plane such that it contains the straight line x-1 2 = y-3 3 = z+2 1 and is perpendicular to the plane x + 2y + 3z = 4 . Let P₁ be the plane which passes through the po 2026Let L be the straight line joining the points P(1, 2, -1) and Q(2, 3, 1) . Let S be the foot of the perpendicular drawn from the point R(4, -1, 5) to the line L . Another line pass 2026Let the foot of perpendicular from the point ( , 2, 3) on the line x-4 1 = y-9 2 = z-5 1 be the point (1, , 2) . Then the distance between the lines x-1 2 = y-2 3 = z+4 6 and x- 2 2026Let a line L₁ pass through the origin and be perpendicular to the lines L₂: r = (3+t) i + (2t-1) j + (2t+4) k and L₃: r = (3+2s) i + (3+2s) j + (2+s) k , t, s R . If (a, b, c) , a 2026The shortest distance between the lines x-4 1 = y-3 2 = z-2 -3 and x+2 2 = y-6 4 = z-5 -5 is : 2026Let the image of the point P(0, -5, 0) in the line x-1 2 = y 1 = z+1 -2 be the point R and the image of the point Q (0, -1 2 , 0 ) in the line x-1 -1 = y+9 4 = z+1 1 be the point S 2026Let the image of the point P(1, 6, a) in the line L: x 1 = y-1 2 = z-a+1 b , b > 0 , be ( a 3 , 0, a+c ) . If S( , , ) , > 0 , is the point on L such that the distance of S from th 2026Let a line L be perpendicular to both the lines L₁: x+1 3 = y+3 5 = z+5 7 and L₂: x-2 1 = y-4 4 = z-6 7 . If is the acute angle between the lines L and L₃: x - 8 7 2 = y - 4 7 1 = 2026 Full Three Dimensional Geometry list All JEE Main PYQs