Advertisements
Advertisements
Question
If a line makes angles 90°, 135°, 45° with the x, y and z axes respectively, find its direction cosines.
Advertisements
Solution 1
A-line makes 90° and 135°, 45°with x, y and z axes, respectively.
Therefore, Direction cosines of the line are cos 90°, cos135°, and cos45°
⇒ Direction cosines of the line are 0, `-(1)/sqrt(2),(1)/sqrt(2)`
Solution 2
Let the direction cosines of the line be l, m and n.
a = 90°, b = 135°, c = 45°
Now,
l = cos a = cos 90° = 0
m = cos b = cos 135° = `-1/sqrt2`
n = cos c = cos 45° = `1/sqrt2`
direction cosines of a line = `0, -1/sqrt2, 1/sqrt2`
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 angle between the lines whose direction ratios are 4, –3, 5 and 3, 4, 5.
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 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.
Find the angle between the lines whose direction cosines are given by the equations
(i) l + m + n = 0 and l2 + m2 − n2 = 0
Find the angle between the lines whose direction cosines are given by the equations
l + 2m + 3n = 0 and 3lm − 4ln + mn = 0
Write the inclination of a line with Z-axis, if its direction ratios are proportional to 0, 1, −1.
Answer each of the following questions in one word or one sentence or as per exact requirement of the question:
Write the distance of a point P(a, b, c) from 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.
A rectangular parallelopiped is formed by planes drawn through the points (5, 7, 9) and (2, 3, 7) parallel to the coordinate planes. The length of an edge of this rectangular parallelopiped is
The xy-plane divides the line joining the points (−1, 3, 4) and (2, −5, 6)
If P (3, 2, −4), Q (5, 4, −6) and R (9, 8, −10) are collinear, then R divides PQ in the ratio
Find the equation of the lines passing through the point (2, 1, 3) and perpendicular to the lines
Verify whether the following ratios are direction cosines of some vector or not
`1/5, 3/5, 4/5`
Verify whether the following ratios are direction cosines of some vector or not
`1/sqrt(2), 1/2, 1/2`
Verify whether the following ratios are direction cosines of some vector or not
`4/3, 0, 3/4`
Find the direction cosines of a vector whose direction ratios are
1, 2, 3
Find the direction cosines of a vector whose direction ratios are
`1/sqrt(2), 1/2, 1/2`
Find the direction cosines and direction ratios for the following vector
`hat"i" - hat"k"`
If `1/2, 1/sqrt(2), "a"` are the direction cosines of some vector, then find a
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 a line makes an angle of 30°, 60°, 90° with the positive direction of x, y, z-axes, respectively, then find its direction cosines.
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 ______.
If a line makes angles `pi/2, 3/4 pi` and `pi/4` with x, y, z axis, respectively, then its direction cosines are ______.
If a line makes angles α, β, γ with the positive directions of the coordinate axes, then the value of sin2α + sin2β + sin2γ is ______.
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
O is the origin and A is (a, b, c). Find the direction cosines of the line OA and the equation of plane through A at right angle to OA.
If the directions cosines of a line are k,k,k, then ______.
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.
If a line has the direction ratio – 18, 12, – 4, then what are its direction cosine.
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 ______.
The projections of a vector on the three coordinate axis are 6, –3, 2 respectively. The direction cosines of the vector are ______.
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.
In the relation between direction ratios and direction cosines, what does the sign \[\pm\] in \[l=\pm\frac{a}{\sqrt{a^2+b^2+c^2}}\] depend on?
