Advertisements
Advertisements
प्रश्न
By computing the shortest distance, determine whether following lines intersect each other.
`bar"r" = (hat"i" - hat"j") + lambda(2hat"i" + hat"k") and bar"r" = (2hat"i" - hat"j") + mu(hat"i" + hat"j" - hat"k")`
उत्तर
The shortest distance between the lines
`bar"r" = bar"a"_1 + lambdabar"b"_1 and bar"r" = bar"a"_2 + mubar"b"_2` is given by
Here, `bar"a"_1 = hat"i" - hat"j", bar"a"_2 = 2hat"i" - hat"j", bar"b"_1 =2hat"i" + hat"k", bar"b"_2 = hat"i" + hat"j" - hat"k"`.
`bara_2 - bara_1 = (2hati-hatj) - (hati-hatj)`
∴ `bara_2 - bara_1 = hati`
∴ `bar"b"_1 xx bar"b"_2 = |(hat"i",hatj, hat"k"),(2, 0, 1),(1, 1, -1)|`
= `(0 - 1)hat"i" - (-2 - 1)hat"j" + (2 - 0)hat"k"`
= `-hat"i" + 3hat"j" + 2hat"k"`
and
∴ `(bar"a"_2 - bar"a"_1).(bar"b"_1 xx bar"b"_2) = hat"i".(-hat"i" + 3hat"j" + 2hat"k")`
= 1(– 1) + 0(3) + 0(2)
= – 1
and
`|bar"b"_1 xx bar"b"_2| = sqrt((-1)^2 + 3^2 + 2^2)`
= `sqrt(1 + 9 + 4)`
= `sqrt(14)`
d = `|((bar"a"_2 - bar"a"_1).(bar"b"_1 xx bar"b"_2))/|bar"b"_1 xx bar"b"_2||`.
∴ the shortest distance between the given lines
= `|(-1)/sqrt(14)|`
= `(1)/sqrt(14)"unit"`
Hence, the given line do not intersect.
APPEARS IN
संबंधित प्रश्न
Find the shortest distance between the lines `barr = (4hati - hatj) + λ(hati + 2hatj - 3hatk)` and `barr = (hati - hatj + 2hatk) + μ(hati + 4hatj - 5hatk)`
Find the shortest distance between the lines `(x + 1)/(7) = (y + 1)/(-6) = (z + 1)/(1) and (x - 3)/(1) = (y - 5)/(-2) = (z - 7)/(1)`
By computing the shortest distance, determine whether following lines intersect each other.
`(x - 5)/(4) = (y -7)/(-5) = (z + 3)/(-5) and (x - 8)/(7) = (y - 7)/(1) = (z - 5)/(3)`
By computing the shortest distance determine whether the following lines intersect each other: `(x -5)/(4) = (y - 7)/(5) = (z + 3)/(5)` and x – 6 = y – 8 = z + 2.
Find the direction cosines of the lines `bar"r" = (-2hat"i" + 5/2hat"j" - hat"k") + lambda(2hat"i" + 3hat"j")`.
Choose correct alternatives :
The shortest distance between the lines `vecr = (hati + 2hatj + hatk) + lambda(hati - hatj + hatk) and vecr = (2hati - hatj - hatk) + μ(2hati + hatj + 2hatk)` is ______.
The direction ratios of the line 3x + 1 = 6y – 2 = 1 – z are ______.
Choose correct alternatives :
If the planes `bar"r".(2hat"i" - lambdahat"j" + hat"k") = 3 and bar"r".(4hat"i" - hat"j" + muhat"k") = 5` are parallel, then the values of λ and μ are respectively
Find the distance between the parallel lines `x/2 = y/(-1) = z/2` and `(x - 1)/2 = (y - 1)/(-1) = (z - 1)/2`
If the line `x/(-3) = "y"/2 = "z"/5` is perpendicular to the line `(x - 2)/2 = ("y - 1")/"k" = ("z - 3")/("k + 11")`, then the value of k is ______.
If the lines `(x + 1)/"k" = ("y + 3")/(- 1) = ("z" - 4)/1` and `(x + 10)/(-1) = ("y + 1")/(-3) = ("z - 1")/4` intersect each other, then the value of k is ______.
The shortest distance between the skew lines `overliner= (6hati + 2hatj + 2hatk) + t(hati - 2hatj + 2hatk)` and `overliner = (-4hati - hatk) + s(3hati - 2hatj - 2hatk)` is ______
Find the distance between the parallel lines `x/2 = y/-1 = z/2` and `(x - 1)/2 = (y - 1)/-1 = (z - 1)/2`
The distance between the lines `x/2 = y/(-1) = z/2` and `(x - 1)/2 = (y - 1)/(-1) = (z - 1)/2` is ______.
If the straight lines `(x - 1)/k = (y - 2)/2 = (z - 3)/3` and `(x - 2)/3 = (y - 3)/k = (z - 1)/2` intersect at a point, then the integer k is equal to ______.
Find the shortest distance between the lines `barr = (4hati - hatj) + λ(hati + 2hatj - 3hatk)` and `barr = (hati - hatj -2hatk) + μ(hati + 4hatj - 5hatk)`
