Advertisements
Advertisements
प्रश्न
Show that the lines `vec"r" = (6hat"i" + hat"j" + 2hat"k") + "s"(hat"i" + 2hat"j" - 3hat"k")` and `vec"r" = (3hat"i" + 2hat"j" - 2hat"k") + "t"(2hat"i" + 4hat"j" - 5hat"k")` are skew lines and hence find the shortest distance between them
उत्तर
`vec"r" = vec"a" + "s"vec"b"`, `vec"r" = vec"c" + "s"vec"d"`
`vec"a" = 6hat"i" + hat"j" + 2hat"k"`, `vec"b" = hat"i" + 2hat"j" - 3hat"k"`
`vec"c" = 3hat"i" + 2hat"j" - 2hat"k"`, `vec"d" = 2hat"i" + 4hat"j" - 5hat"k"`
`vec"b"` is not a scalar multiple of `vec"d"`
∴ They are not parallel.
∴ The given lines are skew lines.
The shortest distance δ = `(|(vec"c" - vec"a")*(vec"b" xx vec"d")|)/(|vec"b" xx vec"d"|)`
`vec"b" xx vec"d" = |(hat"i", hat"j", hat"k"),(1, 2, -3),(2, 4, 5)|`
= `"i"(- 10 + 12) - hat"j"(- 5 + 6) + hat"k"(4 - 4)`
= `2hat"i" - hat"j"`
`|vec"b" xx vec"d"| = sqrt(2^2 + 1^2) = sqrt(5)`
`vec"c" - vec"a" = 3hat"i" + 2hat"j" - 2hat"k" - 6hat"i" - hat"j" - 2hat"k"`
= `3hat"i" + hat"j" - 4hat"k"`
`("c" - "a")*(vec"b" xx vec"d") = (-3hat"i" + hat"j" - 4hat"k")*(2hat"i" - hat"j")`
= – 6 – 1 + 0
= – 7
δ = `(|(vec"c" - vec"a")*(vec"b" xx vec"d")|)/(|vec"b" xx vec"d"|)`
= `|-7|/sqrt(5)`
= `7/sqrt(5)` units
APPEARS IN
संबंधित प्रश्न
Find the non-parametric form of vector equation and Cartesian equations of the straight line passing through the point with position vector `4hat"i" + 3hat"j" - 7hat"k"` and parallel to the vector `2hat"i" - 6hat"j" + 7hat"k"`
Find the parametric form of vector equation and Cartesian equations of the straight line passing through the point (– 2, 3, 4) and parallel to the straight line `(x - 1)/(-4) = (y + 3)/5 = (8 - z)/6`
Find the acute angle between the following lines.
`(x + 4)/3 = (y - 7)/4 = (z + 5)/5, vec"r" = 4hat"k" + "t"(2hat"i" + hat"j" + hat"k")`
f the straight line joining the points (2, 1, 4) and (a – 1, 4, – 1) is parallel to the line joining the points (0, 2, b – 1) and (5, 3, – 2) find the values of a and b
Show that the points (2, 3, 4), (– 1, 4, 5) and (8, 1, 2) are collinear
Find the parametric form of vector equation and Cartesian equations of straight line passing through (5, 2, 8) and is perpendicular to the straight lines `vec"r" = (hat"i" + hat"j" - hat"k") + "s"(2hat"i" - 2hat"j" + hat"k")` and `vec"r" = (2hat"i" - hat"j" - 3hat"k") + "t"(hat"i" + 2hat"j" + 2hat"k")`
If the two lines `(x - 1)/2 = (y + 1)/3 = (z - 1)/4` and `(x - 3)/1 = (y - "m")/2` = z intersect at a point, find the value of m
Show that the lines `(x - 3)/3 = (y - 3)/(-1), z - 1` = 0 and `(x - 6)/2 = (z - 1)/3, y - 2` = 0 intersect. Aslo find the point of intersection
Show that the straight lines x + 1 = 2y = – 12z and x = y + 2 = 6z – 6 are skew and hence find the shortest distance between them
Find the parametric form of vector equation of the straight line passing through (−1, 2, 1) and parallel to the straight line `vec"r" = (2hat"i" + 3hat"j" - hat"k") + "t"(hat"i" - 2hat"j" + hat"k")` and hence find the shortest distance between the lines
Find the foot of the perpendicular drawn: from the point (5, 4, 2) to the line `(x + 1)/2 = (y - 3)/3 = (z - 1)/(-1)`. Also, find the equation of the perpendicular
Choose the correct alternative:
If `[vec"a", vec"b", vec"c"]` = 1, then the value of `(vec"a"*(vec"b" xx vec"c"))/((vec"c" xx vec"a")*vec"b") + (vec"b"*(vec"c" xx vec"a"))/((vec"a" xx vec"b")*vec"c") + (vec"c"*(vec"a" xx vec"b"))/((vec"c" xx vec"b")*vec"a")` is
Choose the correct alternative:
If `vec"a", vec"b", vec"c"` are non-coplanar, non-zero vectors `[vec"a", vec"b", vec"c"]` = 3, then `{[[vec"a" xx vec"b", vec"b" xx vec"c", vec"c" xx vec"a"]]}^2` is equal to
Choose the correct alternative:
I`vec"a" xx (vec"b" xx vec"c") = (vec"a" xx vec"b") xx vec"c"`, where `vec"a", vec"b", vec"c"` are any three vectors such that `vec"b"*vec"c" ≠ 0` and `vec"a"*vec"b" ≠ 0`, then `vec"a"` and `vec"c"` are
