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
Find the direction cosines of the line perpendicular to the lines whose direction ratios are -2, 1,-1 and -3, - 4, 1
Find the direction cosines of the line
`(x+2)/2=(2y-5)/3; z=-1`
Write the direction ratios of the following line :
`x = −3, (y−4)/3 =( 2 −z)/1`
Find the angle between the lines whose direction ratios are 4, –3, 5 and 3, 4, 5.
Show that the points (2, 3, 4), (−1, −2, 1), (5, 8, 7) are collinear.
If the lines `(x-1)/(-3) = (y -2)/(2k) = (z-3)/2 and (x-1)/(3k) = (y-1)/1 = (z -6)/(-5)` are perpendicular, find the value of k.
If a line makes angles of 90°, 60° and 30° with the positive direction of x, y, and z-axis respectively, find its direction cosines
Find the angle between the vectors whose direction cosines are proportional to 2, 3, −6 and 3, −4, 5.
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 direction cosines of the lines, connected by the relations: l + m +n = 0 and 2lm + 2ln − mn= 0.
What are the direction cosines of Y-axis?
Write the distance of the point (3, −5, 12) from X-axis?
A line makes an angle of 60° with each of X-axis and Y-axis. Find the acute angle made by the line with Z-axis.
If a line makes angles α, β and γ with the coordinate axes, find the value of cos2α + cos2β + cos2γ.
Write the inclination of a line with Z-axis, if its direction ratios are proportional to 0, 1, −1.
Write the angle between the lines whose direction ratios are proportional to 1, −2, 1 and 4, 3, 2.
Write the distance of the point P (x, y, z) from XOY plane.
Find the distance of the point (2, 3, 4) from the x-axis.
If a line has direction ratios proportional to 2, −1, −2, then what are its direction consines?
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.
If O is the origin, OP = 3 with direction ratios proportional to −1, 2, −2 then the coordinates of P are
Find the equation of the lines passing through the point (2, 1, 3) and perpendicular to the lines
Find the direction cosines of the line joining the points P(4,3,-5) and Q(-2,1,-8) .
Find the direction cosines and direction ratios for the following vector
`3hat"i" - 3hat"k" + 4hat"j"`
If (a, a + b, a + b + c) is one set of direction ratios of the line joining (1, 0, 0) and (0, 1, 0), then find a set of values of a, b, c
If `vec"a" = 2hat"i" + 3hat"j" - 4hat"k", vec"b" = 3hat"i" - 4hat"j" - 5hat"k"`, and `vec"c" = -3hat"i" + 2hat"j" + 3hat"k"`, find the magnitude and direction cosines of `3vec"a"- 2vec"b"+ 5vec"c"`
If the direction ratios of a line are 1, 1, 2, find the direction cosines of the line.
Find the direction cosines of the line passing through the points P(2, 3, 5) and Q(–1, 2, 4).
A line makes equal angles with co-ordinate axis. Direction cosines of this line are ______.
If a line makes an angle of `pi/4` with each of y and z-axis, then the angle which it makes with x-axis is ______.
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")`
Find the equations of the two lines through the origin which intersect the line `(x - 3)/2 = (y - 3)/1 = z/1` at angles of `pi/3` each.
What will be the value of 'P' so that the lines `(1 - x)/3 = (7y - 14)/(2P) = (z - 3)/2` and `(7 - 7x)/(3P) = (y - 5)/1 = (6 - z)/5` at right angles.
The Cartesian equation of a line AB is: `(2x - 1)/2 = (y + 2)/2 = (z - 3)/3`. Find the direction cosines of a line parallel to line AB.
If the equation of a line is x = ay + b, z = cy + d, then find the direction ratios of the line and a point on the line.
Find the coordinates of the image of the point (1, 6, 3) with respect to the line `vecr = (hatj + 2hatk) + λ(hati + 2hatj + 3hatk)`; where 'λ' is a scalar. Also, find the distance of the image from the y – axis.
