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 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 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 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`
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 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")`
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)/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:
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 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 line `(x - )/3 = (y - 1)/(-5) = (x + 2)/2` lies in the plane x + 3y – αz + ß = 0 then (α + ß) 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 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 ______.
