Advertisements
Advertisements
प्रश्न
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
Advertisements
उत्तर
Let `vec"a" = 5hat"i" + 7hat"j" - 3hat"k"`
`vec"b" = 4hat"i" + 4hat"j" - 5hat"k"`
`vec"c" = 8hat"i" + 4hat"j" + 5hat"k"`
`vec"d" = 7hat"i" + hat"j" + 3hat"k"`
We know that given two lines are coplanar if
`(vec"c" - vec"a")*(vec"b" xx vec"d")` = 0 ......(1)
`vec"b" xx vec"d" = |(vec"i", vec"j", vec"k"),(4, 4, -5),(7, 1, 3)|`
= `vec"i"(12 + 5) - vec"j"(12 + 35) + vec"k"(4 - 28)`
`vec"b" xx vec"d" = 17hat"i" - 47hat"j" - 24hat"k"`
`vec"c" - vec"a" = (8hat"i" + 4hat"j" + 5hat"k") - (5hat"i" + 7hat"j" - 3hat"k") = 3hat"i" - 3hat"j" + 8hat"k"`
(1) ⇒ `(3hat"i" - 3hat"j" + 8hat"k")*(17hat"i" - 47hat"j" - 24hat"k")` = 51 + 141 – 192 = 0
∴ The two given lines are colpanar so, the non-parametric vector equation is
`(vec"r" - vec"a")*(vec"b" xx vec"d")` = 0
`vec"r"*(vec"b" xx vec"d") = vec"a"*(vec"b" xx vec"d")`
`vec"r"*(17vec"i" - 47vec"j" - 24vec"k") = (5vec"i" + 7vec"j" - 3vec"k")(17vec"i" - 47vec"j" - 24vec"k")`
`vec"r"*(17vec"i" - 47vec"j" - 24vec"k")` = 85 – 329 + 72
⇒ `vec"r"*(17vec"i" - 47vec"j" - 24vec"k")` = – 172
APPEARS IN
संबंधित प्रश्न
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"`
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
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), (9, 3, 6) and perpendicular to the plane 2x + 6y + 6z = 9
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)
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 `vec"a", vec"b", vec"c"` are three non-coplanar vectors such that `vec"a" xx (vec"b" xx vec"c") = (vec"b" + vec"c")/sqrt(2)` then the angle between `vec"a"` and `vec"b"` 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 distance between the planes x + 2y + 3z + 7 = 0 and 2x + 4y + 6z + 7 = 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
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 `(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 ______.
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 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 ______.
