Advertisements
Advertisements
प्रश्न
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
Advertisements
उत्तर
`(x - 3)/3 = (y - 3)/(-1), z - 1` = 0 ⇒ z = 1
`(x - 6)/2 = (z - 1)/3, y - 2` = 0 ⇒ y = 2
(x1, y1, z1) = (3, 3, 1) and (x2, y2, z2) = (6, 2, 1)
(b1, b2, b3) = (3, –1, 0) and (d1, d2, d3) = (2, 0, 3)
Condition for intersection of two lines
`|(x_2 - x_1, y_2 - y_1, z_2 - z_1),("b"_1, "b"_2, "b"_3),("d"_1, "d"_2, "d"_3)|` = 0
`|(3, -1, 0),(3, 1, 0),(2, 1, 3)|` = 0 Since (R1 = R2)
∴ Given two lines are intersecting lines.
Any point on the first time
`(x - 3)/3 = (y - 3)/(-1) = lambda` and z = 1
`(3lambda + 3, -lambda + 3, 1)`
Any point on the Second line
`(x - 6)/2 = (z - 1)/3 = mu` and y = 2
`(2mu + 6, 2, 3mu + 1)`
∴ `3mu + 1` = 1
`3mu` = 0
`mu` = 0
`-lambda + 3` = 2
`-lambda` = – 1
`lamda` = 1
∴ The required point of intersection is (6, 2, 1)
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 acute angle between the following lines.
`vec"r" = (4hat"i" - hat"j") + "t"(hat"i" + 2hat"j" - 2hat"k")`
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")`
Find the acute angle between the following lines.
2x = 3y = – z and 6x = – y = – 4z
The vertices of ΔABC are A(7, 2, 1), 5(6, 0, 3), and C(4, 2, 4). Find ∠ABC
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")`
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
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
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
Choose the correct alternative:
The vector equation `vec"r" = (hat"i" - hat"j" - hat"k") + "t"(6hat"i" - hat"k")` represents a straight line passing through the points
