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 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 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)
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:
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 `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 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:
Distance from the origin to the plane 3x – 6y + 2z + 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
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 ______.
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 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 ______.
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 ______.
The point in which the join of (–9, 4, 5) and (11, 0, –1) is met by the perpendicular from the origin is ______.
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 ______.
