Advertisements
Advertisements
Question
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")`
Solution
`vec"a" = 6hat"i" - hat"j" + hat"k"`
`vec"b" = -hat"i" + 2hat"j" + hat"k"`
`vec"c" = -5hat"i" - 4hat"j" - 5hat"k"`
`vec"b" xx vec"c" = |(hat"i", hat"j", hat"k"),(-1, 2, 1),(-5, -4, -5)|`
= `hat"i"(- 10 + 4) - hat"j"(5 + 5) + hat"k"(4 + 10)`
= `-6hat"i" - 10hat"j" + 14hat"k"`
= `-2(3hat"i" + 5hat"j" - 7hat"k")`
Non-parametric vector equation:
`(vec"r" - vec"a")*(vec"b" xx vec"c") = vec0`
`[vec"r" - (6hat"i" - hat"j" + hat"k")]*(3hat"i" + 5hat"j" - 7hat"k") = vec0`
`vec"r"*(3hat"i" + 5hat"j" - 7hat"k")` = 18 – 5 – 7 = 6
`vec"r"*(3hat"i" + 5hat"j" - 7hat"k")` = 6
Cartesian equation:
3x + 5y – 7z = 6
3x + 5y – 7z – 6 = 0
APPEARS IN
RELATED QUESTIONS
A plane passes through the point (− 1, 1, 2) and the normal to the plane of magnitude `3sqrt(3)` makes equal acute angles with the coordinate axes. Find the equation of the plane
Find the intercepts cut off by the plane `vec"r"*(6hat"i" + 45hat"j" - 3hat"k")` = 12 on the coordinate axes
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 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`
Show that the straight lines `vec"r" = (5hat"i" + 7hat"j" - 3hat"k") + "s"(4hat"i" + 4hat"j" - 5hat"k")` and `vec"r"(8hat"i" + 4hat"j" + 5hat"k") + "t"(7hat"i" + hat"j" + 3hat"k")` are coplanar. Find the vector equation of the plane in which they lie
If the straight lines `(x - 1)/1 - (y - 2)/2 = (z - 3)/"m"^2` and `(x - 3)/5 = (y - 2)/"m"^2 = (z - 1)/2` are coplanar, find the distinct real values of m
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:
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:
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
Let `(x - 2)/3 = (y + 1)/(-2) = (z + 3)/(-1)` lie on the plane px – qy + z = 5, for p, q ∈ R. The shortest distance of the plane from the origin 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 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 ______.
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 ______.
