Advertisements
Advertisements
प्रश्न
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
Advertisements
उत्तर
`vec"a" = hat"i" - hat"j" + 3hat"k"`
`vec"b" = 2hat"i" - hat"j" + 4hat"k"`
`vec"c" = hat"i" + 2hat"j" + hat"k"`
Parametric form of vector equation
`vec"r" = vec"a" + "s"vec"b" + "t"vec"c"`
`vec"r" = (hat"i" - hat"j" + 3hat"k") + "s"(2hat"j" - hat"i" + 4hat"k") + "t"(hat"i" + 2hat"j" + hat"k")`
`vec"b" xx vec"c" = |(hat"i", hat"j", hat"k"),(2, -1, 4),(1, 2, 1)|`
= `hat"i"(- 1 - 8) - hat"j"(2 - 4) + hat"k"(4 + 1)`
= `9hat"i" + 2hat"j" + 5hat"k"`
= `-(9hat"i" - 2hat"j" - 5hat"k")`
Non-parametric form of vector equation:
`(vec"r" - vec"a")*(vec"b" xx vec"c") = vec0`
`[vec"r" - (hat"i" - hat"j" + 3hat"k")]*(-(9hat"i" - 2hat"j" - 5hat"k")) = vec0`
`vec"r"*(+9hat"i" - 2hat"j" - 5hat"k")` = + 9 + 2 – 15
`[vec"r"*(9hat"i" - 2hat"j" - 5hat"k")]` = – 4
Cartesian equation
9x – 2y – 5z = – 4
9x – 2y – 5z + 4 = 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 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)`
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:
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:
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"` 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 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:
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:
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
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
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 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 ______.
