Advertisements
Advertisements
प्रश्न
Show that the points (2, 3, 4), (−1, −2, 1), (5, 8, 7) are collinear.
Advertisements
उत्तर
Let A = (2, 3, 4), B = (-1, -2, 1) and C = (5, 8, 7)
Direction ratio of AB are < (-1 - 2), (- 2 - 3), (1 - 4) >
⇒ i.e., < -3, -5, -3 >
Direction ratio of AC are < (5 - 2), (8 - 3), (7 - 4) >
⇒ i.e., < 3, 5, 3 >
It is clear that the direction ratios of AB and AC are proportional.
Hence, AB and AC are parallel, but these have a point A in common.
Therefore A, Band Care collinear.
APPEARS IN
संबंधित प्रश्न
Write the direction ratios of the following line :
`x = −3, (y−4)/3 =( 2 −z)/1`
If a line makes angles 90°, 135°, 45° with the X, Y, and Z axes respectively, then its direction cosines are _______.
(A) `0,1/sqrt2,-1/sqrt2`
(B) `0,-1/sqrt2,-1/sqrt2`
(C) `1,1/sqrt2,1/sqrt2`
(D) `0,-1/sqrt2,1/sqrt2`
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 has direction ratios 2, −1, −2, determine its direction cosines.
Using direction ratios show that the points A (2, 3, −4), B (1, −2, 3) and C (3, 8, −11) are collinear.
Find the angle between the vectors with direction ratios proportional to 1, −2, 1 and 4, 3, 2.
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.
Define direction cosines of a directed line.
Write the distances of the point (7, −2, 3) from XY, YZ and XZ-planes.
Write the distance of the point (3, −5, 12) from X-axis?
If a line makes angles α, β and γ with the coordinate axes, find the value of cos2α + cos2β + cos2γ.
Write the ratio in which the line segment joining (a, b, c) and (−a, −c, −b) is divided by the xy-plane.
Write the inclination of a line with Z-axis, if its direction ratios are proportional to 0, 1, −1.
Write the distance of the point P (x, y, z) from XOY plane.
If a line has direction ratios proportional to 2, −1, −2, then what are its direction consines?
If a unit vector `vec a` makes an angle \[\frac{\pi}{3} \text{ with } \hat{i} , \frac{\pi}{4} \text{ with } \hat{j}\] and an acute angle θ with \[\hat{ k} \] ,then find the value of θ.
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 x-axis (except the origin),
The distance of the point P (a, b, c) from the x-axis is
The angle between the two diagonals of a cube is
If a line makes angles 90°, 135°, 45° with the x, y and z axes respectively, find its direction cosines.
Verify whether the following ratios are direction cosines of some vector or not
`1/sqrt(2), 1/2, 1/2`
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 `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 `vec"a", vec"b", vec"c"`
Choose the correct alternative:
The unit vector parallel to the resultant of the vectors `hat"i" + hat"j" - hat"k"` and `hat"i" - 2hat"j" + hat"k"` is
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).
P is a point on the line segment joining the points (3, 2, –1) and (6, 2, –2). If x co-ordinate of P is 5, then its y co-ordinate is ______.
If a line makes angles α, β, γ with the positive directions of the coordinate axes, then the value of sin2α + sin2β + sin2γ is ______.
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")`
The direction cosines of vector `(2hat"i" + 2hat"j" - hat"k")` are ______.
A line passes through the points (6, –7, –1) and (2, –3, 1). The direction cosines of the line so directed that the angle made by it with positive direction of x-axis is acute, are ______.
Equation of a line passing through point (1, 2, 3) and equally inclined to the coordinate axis, is ______.
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.
