Advertisements
Advertisements
प्रश्न
If a line makes angles 90°, 135°, 45° with the X-, Y- and Z-axes respectively, then find its direction cosines.
उत्तर
Let l, m, n be the direction cosines of the line.
Then l = cos α, m = cos β, n = cos γ
Here, α = 90°, β = 135°, γ = 45°
∴ l = cos 90° = 0
m = cos 135° = cos (180° - 45°) = - cos 45°
`= - 1/sqrt2` and n = cos 45° = `1/sqrt2`
∴ the direction cosines of the line are 0, `- 1/sqrt2, 1/sqrt2`
APPEARS IN
संबंधित प्रश्न
Show that the sum of the length of projections of `"p"hat"i" + "q"hat"j" + "r"hat"k"` on the coordinate axes, where p = 2, q = 3 and r = 4 is 9.
If `bar"p", bar"q"` and `bar"r"` are unit vectors, find `bar"p".bar"r".`
The direction ratios of `bar"AB"` are −2, 2, 1. If A = (4, 1, 5) and l(AB) = 6 units, find B.
Find a unit vector perpendicular to the vectors `hat"j" + 2hat"k"` and `hat"i" + hat"j"`.
If `bar"a" = 2hat"i" + hat"j" - 3hat"k"` and `bar"b" = hat"i" - 2hat"j" + hat"k"`, find a vector of magnitude 5 perpendicular to both `bar"a"` and `bar"b"`.
Find `|bar"u" xx bar"v"|` if `|bar"u"| = 10, |bar"v"| = 2, bar"u".bar"v" = 12`
Find the area of the parallelogram whose adjacent sides are `bar"a" = 2hat"i" - 2hat"j" + hat"k"` and `bar"b" = hat"i" - 3hat"j" - 3hat"k"`
Show that vector area of a parallelogram ABCD is `1/2 (bar"AC" xx bar"BD")` where AC and BD are its diagonals.
Find the area of parallelogram whose diagonals are determined by the vectors `bar"a" = 3hat"i" - hat"j" - 2hat"k"` and `bar"b" = - hat"i" + 3hat"j" - 3hat"k"`.
If `bar"a" = hat"i" + hat"j" + hat"k" "and" bar"c" = hat"j" - hat"k"`, find `bar"a"` vector `bar"b"` satisfying `bar"a" xx bar"b" = bar"c" "and" bar"a".bar"b" = 3`
Find `bar"a"` if `bar"a" xx hat"i" + 2bar"a" - 5hat"j" = bar"0"`
If `|bar"a".bar"b"| = |bar"a" xx bar"b"|` and `bar"a".bar"b" < 0`, then find the angle between `bar"a" "and" bar"b"`.
Prove, by vector method, that sin (α + β) = sin α . cos β + cos α . sin β
Find the direction ratios of a vector perpendicular to the two lines whose direction ratios are - 2, 1, - 1 and - 3, - 4, 1
Prove that the two vectors whose direction cosines are given by relations al + bm + cn = 0 and fmn + gnl + hlm = 0 are perpendicular, if `"f"/"a" + "g"/"b" + "h"/"c" = 0`
If A(1, 2, 3) and B(4, 5, 6) are two points, then find the foot of the perpendicular from the point B to the line joining the origin and the point A.
The angle θ between two non-zero vectors `bar("a")` and `bar("b")` is given by cos θ = ______
The value of `hat"i"*(hat"j" xx hat"k") + hat"j"*(hat"i" xx hat"k") + hat"k"*(hat"i" xx hat"j")`.
If `|bar("a")*bar("b")| = |bar("a") xx bar("b")|` and `bar("a")*bar("b") < 0`, then find the angle between `bar("a")` and `bar("b")`
The area of triangle ABC in which c = 8 , b = 3, ∠A = 60° is ______
If the vectors `ahat("i")+hat("j")+hat("k"), hat("i")+bhat("j")+hat("k")` and `hat("i")+hat("j")+chat("k")` are coplanar (a ≠ b ≠ c ≠ 1), then the value of abc - (a + b + c) = ______.
If `veca, vecb, vecc` are vectors such that `[(veca, vecb, vecc)]` = 4, then `[(veca xx vecb, vecb xx vecc, vecc xx veca)]` = ______.
For non zero, non collinear vectors `vecp` and `vecq`, the value of `[(hati, vecp, vecq)]hati + [(hatj, vecp, vecq)]hatj + [(hatk, vecp, vecq)]hatk` is ______.
Let `veca, vecb` and `vecc` be non-coplanar unit vectors equally inclined to one another at an acute angle θ. Then `[(veca, vecb, vecc)]` in terms of θ is equal to ______.
Find two unit vectors each of which is perpendicular to both `baru "and" barv`, where `baru =2hati + hatj - 2hatk, barv =hati + 2hatj - 2hatk `
Find two unit vectors each of which is perpendicular to both
`baru "and" barv, "where" baru = 2hati + hatj - 2hatk, barv = hati + 2hatj - 2hatk`
Find two unit vectors each of which is perpendicular to both `\overline "u" and \overline "v",` where ` \overline "u" = 2hati + hatj - 2hatk, \overline "v" = hati + 2hatj - 2hatk`
If a vector has direction angles 45º and 60º find the third direction angle.
Find two unit vectors each of which is perpendicular to both `baru and barv` where `baru = 2hati +hatj -2hatk, barv = hati +2hatj-2hatk`
Find the direction ratios of a line perpendicular to both the lines whose direction ratios are 3, –2, 1 and 2, 4, –2
Find two unit vectors each of which is perpendicular to both `baru and barv`, where `baru = 2hati + hatj - 2hatk, barv = hati + 2hatj - 2hatk`
Find two unit vectors each of which is perpendicular to both `baru and barv, "where" baru = 2hati + hatj - 2hatk, barv = hati + 2hatj - 2hatk`
If a vector has direction angles 45ºand 60º find the third direction angle.
Find two unit vectors each of which is perpendicular to both `baru and barv , "where" baru = 2hati + hatj - 2hatk, barv = hati + 2hatj -2hatk`
