Advertisements
Advertisements
Question
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"`.
Solution
Let θ be the angle between `bar"a"` and `bar"b"`
∵ `bar"a" xx bar"b" = 2hat"i" + hat"j" + 2hat"k"`
∴ `|bar"a" xx bar"b"| = sqrt(2^2 + 1^2 + 2^2) = sqrt(4 + 1 + 4) = 3`
∴ `|bar"a"||bar"b"|` sin θ = 3 ...(1)
∴ `bar"a".bar"b" = sqrt3`
∴ `|bar"a"||bar"b"| "cos" theta = sqrt3` ....(2)
∴ Dividing (1) by (2), we get
`(|bar"a"||bar"b"| "sin" theta)/(|bar"a"||bar"b"| "cos" theta) = 3/sqrt3`
∴ tan θ = `sqrt3 = tan 60^circ`
∴ θ = 60°
APPEARS IN
RELATED QUESTIONS
If `veca` and `vecb` are two vectors perpendicular to each other, prove that `(veca + vecb)^2 = (veca - vecb)^2`
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.
Find the angle P of the triangle whose vertices are P(0, - 1, - 2), Q(3, 1, 4) and R(5, 7, 1).
If `hat"p", hat"q"` and `hat"r"` are unit vectors `hat"p"+hat "r" = hat "q"`, find `hat"p".hat"q".`
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"`.
Find `|bar"u" xx bar"v"|` if `|bar"u"| = 10, |bar"v"| = 2, bar"u".bar"v" = 12`
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`
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
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")`.
Find the direction ratios of a vector perpendicular to the two lines whose direction ratios are 1, 3, 2 and –1, 1, 2
The area of triangle ABC in which c = 8 , b = 3, ∠A = 60° is ______
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 `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.
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 `\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`
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 `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`
Find two unit vectors each of which is perpendicular to both `baru and barv,` where `baru = 2hati + hatj - 2hatk, barv = hati + 2hatj - 2hatk`
