Advertisements
Advertisements
प्रश्न
Find the parametric vector, non-parametric vector and Cartesian form of the equation of the plane passing through the point (3, 6, – 2), (– 1, – 2, 6) and (6, 4, – 2)
Advertisements
उत्तर
`vec"a" = 3hat"i" + 6hat"j" - 2hat"k"`
`vec"b" = -hat"i" - 2hat"j" + 6hat"k"`
`vec"b" - vec"a" = -4hat"i" - 8hat"j" + 8hat"k"`
`vec"c" - vec"a" = 3hat"i" - 2hat"j"`
`(vec"b" - vec"a") xx (vec"c" - vec"a") = |(hat"i", hat"j", hat"k"),(-4, -8, 8),(3, -2, 0)|`
= `hat"i"(0 + 16) - hat"j"(0 - 24) + hat"k"(8 + 24)`
= `16hat"i" + 24hat"j" + 32hat"k"`
Parametric equation:
`vec"r" = vec"a" + "s"(vec"b" - vec"a") + "t"(vec"c" - vec"a")`
`vec"r" = (3hat"i" + 6hat"j" - 2hat"k") + "s"(-4hat"i" - 8hat"j" + 8hat"k") + "t"(3hat"i" - 2hat"j"), "s", "t" ∈ "R"`
Non-parametric equation:
`(vec"r" - vec"a")*[(vec"b" - vec"a") xx (vec"c" - vec"a")] = vec0`
`(vec"r" - vec"a")*(16hat"i" + 24hat"j" + 32hat"k") = vec0`
`[vec"r"*(16hat"i" + 24hat"j" + 32hat"k")] = (3hat"i" + 6hat"j" - 2hat"k")*(16hat"i" + 24hat"j" + 32hat"k")`
`vec"r"(16hat"i" + 24hat"j" + 32hat"k")` = 128
Cartesian equation:
⇒ `vec"r"(2hat"i" + 3hat"j" + 4hat"k")` = 16
⇒ 2x + 3y + 4z – 16 = 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
Find the intercepts cut off by the plane `vec"r"*(6hat"i" + 45hat"j" - 3hat"k")` = 12 on the coordinate axes
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 parametric form of vector equation, and Cartesian equations of the plane containing the line `vec"r" = (hat"i" - hat"j" + 3hat"k") + "t"(2hat"i" - hat"j" + 4hat"k")` and perpendicular to plane `vec"r"*(hat"i" + 2hat"j" + hat"k")` = 8
If the straight lines `(x - 1)/2 = (y + 1)/lambda = z/2` and `(x + 1)/5 = (y + 1)/2 = z/lambda` are coplanar, find λ and equations of the planes containing these two lines
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 `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:
The angle between the lines `(x - 2)/3 = (y + 1)/(-2)`, z = 2 ad `(x - 1)/1 = (2y + 3)/3 = (z + 5)/2` 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:
The distance between the planes x + 2y + 3z + 7 = 0 and 2x + 4y + 6z + 7 = 0 is
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 ______.
A plane P contains the line x + 2y + 3z + 1 = 0 = x – y – z – 6, and is perpendicular to the plane –2x + y + z + 8 = 0. Then which of the following points lies on P?
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 ______.
Consider a plane 2x + y – 3z = 5 and the point P(–1, 3, 2). A line L has the equation `(x - 2)/3 = (y - 1)/2 = (z - 3)/4`. The co-ordinates of a point Q of the line L such that `vec(PQ)` is parallel to the given plane are (α, β, γ), then the product βγ is ______.
