Advertisements
Advertisements
प्रश्न
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
उत्तर
The required plane passes through the points
`vec"a" = 2vec"i" + 2vec"j" + vec"k"`
`vec"b" = 9vec"i" + 3vec"j" + 6vec"k"`
And parallel to the vector `vec"c" = 2vec"i" + 6vec"j" + 6vec"k"`
`vec"b" - vec"a" = (9vec"i" + 3vec"j" + 6vec"k") - (2vec"i" + 2vec"j" + vec"k")`
= `4vec"i" + vec"j" + 5vec"k"`
Nom-aprametric form of vector equation
`(vec"r" - vec"a")*((vec"b" - vec"a") xx vec"c")` = 0
`(vec"b" - vec"a") xx vec"c" = |(vec"i", vec"j", vec"k"),(7, 1, 5),(, 6, 6)|`
= `vec"i"(6 - 30) - vec"j"(42 - 10) + vec"k"(42 - 2)`
= `- 24hat"i" - 32hat"j" + 40hat"k"`
(1) ⇒ `(vec"r" - vec"a")*(-24hat"i" - 32hat"j" + 40hat"k")` = 0
`vec"r"*(-24hat"i" - 32hat"j" + 40hat"k") = vec"a"*(-24vec"i" - 32vec"j" + 40vec"k")`
`"r"*(24vec"i" - 32vec"j" + 40vec"k") = (2vec"i" + 2vec"j" + vec"k")(-24vec"i" - 32vec"j" + 40vec"k")`
`vec"r"*(-24hat"i" - 32hat"j" + 40hat"k") = - 48 - 64 + 40`
`-8[vec"r"*(3hat"i" + 4hat"j" - 5hat"k")]` = – 72
`vec"r"*(3hat"i" + 4hat"j" - 5hat"k")` = 9
Cartesian equation
`(xvec"i" + yvec"j" + xvec"k")*(3vec"i" + 4vec"j" - 5vec"k")` = 9
`3x + 4y - 5z - 9` = 0
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
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
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), (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 equation of the plane passing through the point (1, − 2, 4) and perpendicular to the plane x + 2y − 3z = 11 and parallel to the line `(x + 7)/3 = (y + 3)/(-1) = z/1`
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:
If `vec"a"` and `vec"b"` are unit vectors such that `[vec"a", vec"b", vec"a" xx vec"b"] = 1/4`, are unit vectors such that `vec"a"` nad `vec"b"` is
Choose the correct alternative:
If `vec"a", vec"b", vec"c"` are three non-coplanar vectors such that `vec"a" xx (vec"b" xx vec"c") = (vec"b" + vec"c")/sqrt(2)` then the angle between `vec"a"` and `vec"b"` 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 `vec"a" = 2hat"i" + 3hat"j" - hat"k", vec"b" = hat"i" + 2hat"j" - 5hat"k", vec"c" = 3hat"i" + 5hat"j" - hat"k"`, then a vector perpendicular to `vec"a"` and lies in the plane containing `vec"b"` and `vec"c"` 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:
The angle between the line `vec"r" = (hat"i" + 2hat"j" - 3hat"k") + "t"(2hat"i" + hat"j" - 2hat"k")` and the plane `vec"r"(hat"i" + hat"j") + 4` = 0 is
Choose the correct alternative:
Distance from the origin to the plane 3x – 6y + 2z + 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
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 equation of a plane containing the line of intersection of the planes 2x – y – 4 = 0 and y + 2z – 4 = 0 and passing through the point (1, 1, 0) 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 ______.
