Advertisements
Advertisements
प्रश्न
A plane passes through the point (− 1, 1, 2) and the normal to the plane of magnitude `3sqrt(3)` makes equal acute angles with the coordinate axes. Find the equation of the plane
उत्तर
Given magnitude = `3sqrt(3)` and `vec"a" = -vec"i" + vec"j" + 2vec"k"`
Then, the normal vector makes equal acute angle with the coordinate axes.
We know that cos2 α + cos2 β + cos2 γ = 1
But α = β = γ
`3cos^2alpha` = 1
⇒ `cos^2alpha = 1/3`
`cosalpha = 1/sqrt(3)`
∴ So `vec"n" = 3sqrt(3) [1/sqrt(3)vec"i" + 1/sqrt(3)vec"j" + 1/sqrt(3)vec"k"]`
`vec"n" = 3vec"i" + 3vec"j" + 3vec"k"`
Vector equation of the plane
`vec"r"*vec"n" = vec"a"*vec"n"`
`vec"r"*vec"n" = (-vec"i" + vec"j" + 2vec"k")(3vec"i" + vec"j" + 3vec"k")`
`vec"r"*(hat"i" + 3hat"j" + 3hat"k") = -3 + 3 + 6`
`vec"r"*(hat"i" + 3hat"j" + 3hat"k")` = 6
or
`vec"r"*(vec"i" + vec"j" + vec"k")` = 2
Cartesian equation of the plane
`(xvec"i" + yvec"j" + zvec"k")*(3vec"i" + 3vec"j" + 3vec"k)` = 6
⇒ 3x + 3y + 3z = 6
or
⇒ x + y + z = 2
APPEARS IN
संबंधित प्रश्न
Find the direction cosines of the normal to the plane 12x + 3y – 4z = 65. Also find the non-parametric form of vector equation of a plane and the length of the perpendicular to the plane from the origin
Find the non-parametric form of vector equation and Cartesian equation of the plane passing through the point (2, 3, 6) and parallel to thestraight lines `(x - 1)/2 = (y + 1)/3 = (x - 3)/1` and `(x + 3)/2 = (y - 3)/(-5) = (z + 1)/(-3)`
Find the parametric form of vector equation, and Cartesian equations of the plane passing through the points (2, 2, 1), (9, 3, 6) and perpendicular to the plane 2x + 6y + 6z = 9
Find the parametric form of vector equation and Cartesian equations of the plane passing through the points (2, 2, 1), (1, – 2, 3) and parallel to the straight line passing through the points (2, 1, – 3) and (– 1, 5, – 8)
Find the non-parametric form of vector equation and Cartesian equations of the plane `vec"r" = (6hat"i" - hat"j" + hat"k") + "s"(-hat"i" + 2hat"j" + hat"k") + "t"(-5hat"i" - 4hat"j" - 5hat"k")`
Choose the correct alternative:
If `vec"a", vec"b", vec"c"` are three unit vectors such that `vec"a"` is perpendicular to `vec"b"`, and is parallel to `vec"c"` then `vec"a" xx (vec"b" xx vec"c")` is equal to
Choose the correct alternative:
The volume of the parallelepiped with its edges represented by the vectors `hat"i" + hat"j", hat"i" + 2hat"j", hat"i" + hat"j" + pihat"k"` is
Choose the correct alternative:
If the volume of the parallelepiped with `vec"a" xx vec"b", vec"b" xx vec"c", vec"c" xx vec"a"` as coterminous edges is 8 cubic units, then the volume of the parallelepiped with `(vec"a" xx vec"b") xx (vec"b" xx vec"c"), (vec"b" xx vec"c") xx (vec"c" xx vec"a")` and `(vec"c" xx vec"a") xx (vec"a" xx vec"b")` as coterminous edges is
Choose the correct alternative:
Consider the vectors `vec"a", vec"b", vec"c", vec"d"` such that `(vec"a" xx vec"b") xx (vec"c" xx vec"d") = vec0`. Let P1 and P2 be the planes determined by the pairs of vectors `vec"a", vec"b"` and `vec'c", vec"d"` respectively. Then the angle between P1 and P2 is
Choose the correct alternative:
If the line `(x - )/3 = (y - 1)/(-5) = (x + 2)/2` lies in the plane x + 3y – αz + ß = 0 then (α + ß) is
Choose the correct alternative:
Distance from the origin to the plane 3x – 6y + 2z + 7 = 0 is
Choose the correct alternative:
The distance between the planes x + 2y + 3z + 7 = 0 and 2x + 4y + 6z + 7 = 0 is
Choose the correct alternative:
If the distance of the point (1, 1, 1) from the origin is half of its distance from the plane x + y + z + k = 0, then the values of k are
Choose the correct alternative:
If the planes `vec"r"(2hat"i" - lambdahat"j" + hatk")` = and `vec"r"(4hat"i" + hat"j" - muhat"k")` = 5 are parallel, then the value of λ and µ are
Let d be the distance between the foot of perpendiculars of the points P(1, 2, –1) and Q(2, –1, 3) on the plane –x + y + z = 1. Then d2 is equal to ______.
Let `(x - 2)/3 = (y + 1)/(-2) = (z + 3)/(-1)` lie on the plane px – qy + z = 5, for p, q ∈ R. The shortest distance of the plane from the origin is ______.
The plane passing through the points (1, 2, 1), (2, 1, 2) and parallel to the line, 2x = 3y, z = 1 also passes through the point ______.
The point in which the join of (–9, 4, 5) and (11, 0, –1) is met by the perpendicular from the origin is ______.
A point moves in such a way that sum of squares of its distances from the co-ordinate axis is 36, then distance of then given point from origin are ______.
