Advertisements
Advertisements
Question
If a line has direction ratios proportional to 2, −1, −2, then what are its direction consines?
Advertisements
Solution
\[\text { If a line has direction ratios proportional to 2, - 1, and - 2, then its direction cosines are } \]
\[ \frac{2}{\sqrt{\left( 2 \right)^2 + \left( - 1 \right) + \left( - 2 \right)^2}}, \frac{- 1}{\sqrt{\left( 2 \right)^2 + \left( - 1 \right) + \left( - 2 \right)^2}}, \frac{- 2}{\sqrt{\left( 2 \right)^2 + \left( - 1 \right) + \left( - 2 \right)^2}} \]
\[ = \frac{2}{3}, - \frac{1}{3}, - \frac{2}{3}\]
\[\text { Thus, the direction cosines are } \frac{2}{3}, - \frac{1}{3}, - \frac{2}{3} .\]
APPEARS IN
RELATED QUESTIONS
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`
Find the direction cosines of a line which makes equal angles with the coordinate axes.
If l1, m1, n1 and l2, m2, n2 are the direction cosines of two mutually perpendicular lines, show that the direction cosines of the line perpendicular to both of these are m1n2 − m2n1, n1l2 − n2l1, l1m2 − l2m1.
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
If a line has direction ratios 2, −1, −2, determine its direction cosines.
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 angle between the vectors with direction ratios proportional to 1, −2, 1 and 4, 3, 2.
Show that the points (2, 3, 4), (−1, −2, 1), (5, 8, 7) are collinear.
Show that the line through the points (1, −1, 2) and (3, 4, −2) is perpendicular to the line through the points (0, 3, 2) and (3, 5, 6).
Find the direction cosines of the lines, connected by the relations: l + m +n = 0 and 2lm + 2ln − mn= 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
Write the distances of the point (7, −2, 3) from XY, YZ and XZ-planes.
Write the ratio in which the line segment joining (a, b, c) and (−a, −c, −b) is divided by the xy-plane.
Write the coordinates of the projection of point P (x, y, z) on XOZ-plane.
Find the distance of the point (2, 3, 4) from the x-axis.
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 xy-plane divides the line joining the points (−1, 3, 4) and (2, −5, 6)
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
Verify whether the following ratios are direction cosines of some vector or not
`1/5, 3/5, 4/5`
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
`5hat"i" - 3hat"j" - 48hat"k"`
Find the direction cosines of the line passing through the points P(2, 3, 5) and Q(–1, 2, 4).
If α, β, γ are the angles that a line makes with the positive direction of x, y, z axis, respectively, then the direction cosines of the line are ______.
A line makes equal angles with co-ordinate axis. Direction cosines of this line are ______.
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.
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 line `vec"r" = 2hat"i" - 3hat"j" - hat"k" + lambda(hat"i" - hat"j" + 2hat"k")` lies in the plane `vec"r".(3hat"i" + hat"j" - hat"k") + 2` = 0.
The d.c's of a line whose direction ratios are 2, 3, –6, are ______.
If two straight lines whose direction cosines are given by the relations l + m – n = 0, 3l2 + m2 + cnl = 0 are parallel, then the positive value of c is ______.
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 angles of 90°, 135° and 45° with the x, y and z axes respectively, then its direction cosines are ______.
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.
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 ______.
