Advertisements
Advertisements
प्रश्न
If `vec"a" = hat"i" + hat"j" + hat"k"` and `vec"c" = hat"j" - hat"k"`. find a vector `vec"b"` satisfying `vec"a" xx vec"b" = vec"c"` and `vec"a"·vec"b"` = 3.
उत्तर
Given, `vec"a" = hat"i" + hat"j" + hat"k"`, `vec"c" = hat"j" - hat"k"`
Let `vec"b" = "x"hat"i"+"y"hat"j"+"z"hat"k"`
Then `vec"a"·vec"b"` = 3 gives
`(hat"i" + hat"j" + hat"k"). ("x"hat"i"+"y"hat"j"+"z"hat"k")`= 3
:. (1)(x) + (1)(y) + (1)(z) = 3
Also, x + y + z = 3 ...(1)
Also, `vec"c" = hat"a"xxhat"b"`
`hat"j" - hat"k" = |(hat"i", hat"j", hat"k"),(1, 1, 1),("x", "y", "z")|`
= (z - y)`hat"i"` - (z - x)`hat"j"` + (y - x)`hat"k"`
= (z - y)`hat"i"` + (x - z)`hat"j"` + (y - x)`hat"k"`
By equality of vectors,
z - y = 0 ...(2)
x - z = 1 ...(3)
y - x = - 1 ...(4)
From (2), y = z
From (3), x = 1 + z
Substituting these values of x and y in (1), we get
1 + z + z + z = 3
`"z" = 2/3`
y = z = `2/3`
x = 1 + z = 1 + `2/3` = `5/3`
`vec"b" 5/3hat"i" + 2/3hat"j" +2/3hat"k"`
i.e, `vec"b" = 1/3(5hat"i" + 2hat"j" + 2hat"k")`
APPEARS IN
संबंधित प्रश्न
Find two unit vectors each of which is perpendicular to both `baru` and `barv` where `baru = 2hati + hatj - 2hatk`, `barv = hati + 2hatj - 2hatk`.
If `veca` and `vecb` are two vectors perpendicular to each other, prove that `(veca + vecb)^2 = (veca - vecb)^2`
If `bar"p", bar"q"` and `bar"r"` are unit vectors, find `bar"p".bar"r".`
If a line makes angles 90°, 135°, 45° with the X-, Y- and Z-axes respectively, then find its direction cosines.
Find a unit vector perpendicular to the vectors `hat"j" + 2hat"k"` and `hat"i" + hat"j"`.
If `bar"a".bar"b" = sqrt3` and `bar"a" xx bar"b" = 2hat"i" + hat"j" + 2hat"k"`, find the angle between `bar"a"` and `bar"b"`.
Find `bar"u".bar"v"` if `|bar"u"| = 2, |bar"v"| = 5, |bar"u" xx bar"v"| = 8`
Find `|bar"u" xx bar"v"|` if `|bar"u"| = 10, |bar"v"| = 2, bar"u".bar"v" = 12`
Prove that `2(bar"a" - bar"b") xx 2(bar"a" + bar"b") = 8(bar"a" xx bar"b")`
If `bar"a" = hat"i" - 2hat"j" + 3hat"k"` , `bar"b" = 4hat"i" - 3hat"j" + hat"k"` , `bar"c" = hat"i" - hat"j" + 2hat"k"` verify that `bar"a"xx(bar"b" + bar"c") = bar"a" xx bar"b" + bar"a" xx bar"c"`
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", bar"b", bar"c", bar"d"` are four distinct vectors such that `bar"a" xx bar"b" = bar"c" xx bar"d"` and `bar"a" xx bar"c" = bar"b" xx bar"d"` prove that `bar"a" - bar"d"` is parallel to `bar"b" - bar"c"`.
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 `|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")`
Find the direction ratios of a vector perpendicular to the two lines whose direction ratios are 1, 3, 2 and –1, 1, 2
Let `bar"a" = 2hat"i" + hat"j" - 2hat"k" and bar"b" = hat"i" + hat"j"`. Let `vec"c"` be a vector such that `|bar"c" - bar"a"| = 3, |(bar"a" xx bar"b") xx bar"c"|` = 3 and the angle between `vec"c" and vec"a" xx vec"b" "be" 30^circ`. Then `vec"a" * vec"c"` is equal to ______.
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 `bar"a"` makes an acute angle with `bar"b", bar"r"*bar"a"` = 0 and `bar"r"xx bar"b" = bar"c" xx bar"b"`, then `bar"r"` = ______.
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`
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 45° and 60° find the third direction angle.
Find two unit vectors each of which is perpendicular to both `baruandbarv, "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 `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`
Find two unit vectors each of which is perpendicular to both `baru and barv,` where `baru = 2hati + hatj - 2hatk, barv = hati + 2hatj - 2hatk`
