Advertisements
Advertisements
प्रश्न
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`
Advertisements
उत्तर
The required plane passing through the point `vec"a" = vec"i" - 2vec"j" + 4vec"k"` and parallel to the plane `vec"b" = vec"i" + 2vec"j" - 3vec"k"` and parallel to the line `vec"c" = 3vec"i" - vec"j" + vec"k"`
`vec"b" xx vec"c" = |(vec"i", vec"j", vec"k"),(1, 2, -3),(3, -1, 1)|`
= `vec"i"(2 - 3) - vec"j"(1 + 9) + vec"k"(-1 - 6)`
`vec"b" xx vec"c" = -vec"i" - 10vec"j" - 7vec"k"`
Non-parametric form of vector equation
`(vec"r" - vec"a")*(vec"b" xx vec"c")` = 0
`(vec"r" - vec"a")*(-vec"i" - 10vec"j" - 7vec"k")` = 0
`vec"r"*(-vec"i" - 10vec"j" - 7vec"k") = vec"a"*(-vec"i" - 10vec"j" - 7vec"k")`
`vec"r"*(-vec"i" - 10vec"j" - 7vec"k") = (vec"i" - 2vec"j" + 4vec"k")(-vec"i" - 10vec"j" - 7vec"k")`
`vec"r"*(-vec"i" - 10vec"j" - 7vec"k")` = – 1 + 20 – 28
`vec"r"*(-vec"i" - 10vec"j" - 7vec"k")` = – 9
or
`vec"r"*(vec"i" + 10vec"j" + 7vec"k")` = 9
Cartesian equation
`(xvec"i" + yvec"j" + zvec"k")*(vec"i" + 10vec"j" + 7vec"k")` = 9
x + 10y + 7z – 9 = 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
Find the vector and Cartesian equation of the plane passing through the point with position vector `2hat"i" + 6hat"j" + 3hat"k"` and normal to the vector `hat"i" + 3hat"j" + 5hat"k"`
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 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 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 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)
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
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 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 distance between the planes x + 2y + 3z + 7 = 0 and 2x + 4y + 6z + 7 = 0 is
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
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 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 ______.
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 ______.
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 ______.
