Advertisements
Advertisements
प्रश्न
Show that the points (2, 3, 4), (−1, −2, 1), (5, 8, 7) are collinear.
Advertisements
उत्तर
\[\text{ Suppose the points are A } \left( 2, 3, 4 \right), B \left( - 1 . - 2, 1 \right) \text { and } C \left( 5, 8, 7 \right) . \]
\[\text { We know that the direction ratios of the line joining the points } \left( x_1 , y_1 , z_1 \right) \text{ and } \left( x_2 , y_2 , z_2 \right) \text{ are } x_2 - x_1 , y_2 - y_1 , z_2 - z_1 . \]
\[\text{ The direction ratios of AB are } \left( - 1 - 2 \right), \left( - 2 - 3 \right), \left( 1 - 4 \right), \text{ i . e }. - 3, - 5, - 3 . \]
\[\text{ The direction ratios of BC are } \left( 5 - \left( - 1 \right) \right), \left( 8 - \left( - 2 \right) \right), \left( 7 - 1 \right), \text { i . e } . 6, 10, 6 . \]
\[ \text{ It can be seen that the direction ratios of BC are - 2 times that of AB, i . e . they are proportional . Therefore, AB is parallel to BC }. \]
\[\text { Since point B is common in both AB and BC, points A, B, and C are collinear } .\]
APPEARS IN
संबंधित प्रश्न
Direction cosines of the line passing through the points A (- 4, 2, 3) and B (1, 3, -2) are.........
Show that the points (2, 3, 4), (−1, −2, 1), (5, 8, 7) are collinear.
Find the Direction Cosines of the Sides of the triangle Whose Vertices Are (3, 5, -4), (-1, 1, 2) and (-5, -5, -2).
Find the vector equation of the plane passing through (1, 2, 3) and perpendicular to the plane `vecr.(hati + 2hatj -5hatk) + 9 = 0`
Find the direction cosines of the sides of the triangle whose vertices are (3, 5, −4), (−1, 1, 2) and (−5, −5, −2).
Show that the line joining the origin to the point (2, 1, 1) is perpendicular to the line determined by the points (3, 5, −1) and (4, 3, −1).
If the coordinates of the points A, B, C, D are (1, 2, 3), (4, 5, 7), (−4, 3, −6) and (2, 9, 2), then find the angle between AB and CD.
Find the angle between the lines whose direction cosines are given by the equations
2l − m + 2n = 0 and mn + nl + lm = 0
Find the angle between the lines whose direction cosines are given by the equations
l + 2m + 3n = 0 and 3lm − 4ln + mn = 0
Find the angle between the lines whose direction cosines are given by the equations
2l + 2m − n = 0, mn + ln + lm = 0
Define direction cosines of a directed line.
What are the direction cosines of Y-axis?
Write the distance of the point (3, −5, 12) from X-axis?
Write the coordinates of the projection of point P (x, y, z) on XOZ-plane.
If a line makes angles 90° and 60° respectively with the positive directions of x and y axes, find the angle which it makes with the positive direction of z-axis.
For every point P (x, y, z) on the xy-plane,
For every point P (x, y, z) on the x-axis (except the origin),
The distance of the point P (a, b, c) from the x-axis is
Ratio in which the xy-plane divides the join of (1, 2, 3) and (4, 2, 1) is
Find the direction cosines of a vector whose direction ratios are
0, 0, 7
Find the direction cosines and direction ratios for the following vector
`3hat"i" + hat"j" + hat"k"`
Find the direction cosines and direction ratios for the following vector
`3hat"i" - 3hat"k" + 4hat"j"`
Find the direction cosines and direction ratios for the following vector
`hat"i" - hat"k"`
If a line makes angles `pi/2, 3/4 pi` and `pi/4` with x, y, z axis, respectively, then its direction cosines are ______.
The vector equation of the line passing through the points (3, 5, 4) and (5, 8, 11) is `vec"r" = 3hat"i" + 5hat"j" + 4hat"k" + lambda(2hat"i" + 3hat"j" + 7hat"k")`
If a variable line in two adjacent positions has direction cosines l, m, n and l + δl, m + δm, n + δn, show that the small angle δθ between the two positions is given by δθ2 = δl2 + δm2 + δn2
The direction cosines of vector `(2hat"i" + 2hat"j" - hat"k")` are ______.
The co-ordinates of the point where the line joining the points (2, –3, 1), (3, –4, –5) cuts the plane 2x + y + z = 7 are ______.
A line in the 3-dimensional space makes an angle θ `(0 < θ ≤ π/2)` with both the x and y axes. Then the set of all values of θ is the interval ______.
Equation of line passing through origin and making 30°, 60° and 90° with x, y, z axes respectively, is ______.
If a line makes an angle α, β and γ with positive direction of the coordinate axes, then the value of sin2α + sin2β + sin2γ will be ______.
A directed line makes angles \[\alpha\], \[\beta\], and \[\gamma\] with the positive x-, y-, and z-axes respectively. What are these angles called?
Which identity must be satisfied by the direction cosines \[(l,m,n)\] of a line?
Any three numbers proportional to the direction cosines of a line are called its:
A line passes through \[P(x_1,y_1,z_1)\] and \[Q(x_2,y_2,z_2)\]. Which is one set of direction ratios of the line?
For points \[P(x_1,y_1,z_1)\] and \[Q(x_2,y_2,z_2)\], which expression gives \[PQ\]?
