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 a parametric form of vector equation of a plane which is at a distance of 7 units from t the origin having 3, – 4, 5 as direction ratios of a normal to it
If a plane meets the co-ordinate axes at A, B, C such that the centroid of the triangle ABC is the point (u, v, w), find the equation of the plane
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
Show that the lines `(x - 2)/1 = (y - 3)/1 = (z - 4)/3` and `(x - 1)/(-3) = (y - 4)/2 = (z - 5)/1` are coplanar. Also, find the plane containing these lines
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 `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:
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
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 the plane passing through the point (1, 2, –3) and perpendicular to the planes 3x + y – 2z = 5 and 2x – 5y – z = 7, 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 ______.
The point in which the join of (–9, 4, 5) and (11, 0, –1) is met by the perpendicular from the origin is ______.
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 ______.
