Advertisements
Advertisements
Question
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
Solution
`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
RELATED QUESTIONS
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 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
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:
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:
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:
If the length of the perpendicular from the origin to the plane 2x + 3y + λz = 1, λ > 0 is `1/5, then the value of λ is
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 ______.
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 ______.
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 ______.
Let (λ, 2, 1) be a point on the plane which passes through the point (4, –2, 2). If the plane is perpendicular to the line joining the points (–2, –21, 29) and (–1, –16, 23), then `(λ/11)^2 - (4λ)/11 - 4` is equal to ______.
