Advertisements
Advertisements
प्रश्न
Find the angle P of the triangle whose vertices are P(0, - 1, - 2), Q(3, 1, 4) and R(5, 7, 1).
उत्तर
The position vectors `bar"p", bar"q",` and `bar"r"` of the points P(0, - 1, - 2), Q(3, 1, 4) and R(5, 7, 1) are
`bar"p" = - hat"j" - 2hat"k"` ,
`bar"q" = 3hat"i" + hat"j" + 4hat"k"`,
`bar"r" = 5hat"i" + 7hat"j" + hat"k"`
∴ `bar"PQ" = bar"q" - bar"p"`
`= (3hat"i" + hat"j" + 4hat"k") - (- hat"j" - 2hat"k")`
`= 3hat"i" + 2hat"j" + 6hat"k"`
and `bar"PR" = bar"r" - bar"p"`
`= (5hat"i" + 7hat"j" + hat"k") - (- hat"j" - 2hat"k")`
`= 5hat"i" + 8hat"j" +3hat"k"`
`= bar"PQ" . bar"PR" = (3hat"i" + 2hat"j" + 6hat"k").(5hat"i" + 8hat"j" +3hat"k")`
`= (3)(5) + (2)(8) + (6)(3)`
= 15 + 16 + 18 = 49
`|bar"PQ"| = sqrt(3^2 + 2^2 + 6^2) = sqrt(9 + 4 + 36) =sqrt49 = 7`
`|bar"PR"| = sqrt(5^2 + 8^2 + 3^2) = sqrt(25 + 64 + 9) = sqrt98 = 7sqrt2`
Using the formula for angle between two vectors,
cos P = `(bar"PQ".bar"PR")/(|bar"PQ"||bar"PR"|)`
`= 49/(7 xx 7sqrt2) = 1/sqrt2 = "cos" 45^circ`
∴ P = 45°.
APPEARS IN
संबंधित प्रश्न
If `veca` and `vecb` are two vectors perpendicular to each other, prove that `(veca + vecb)^2 = (veca - vecb)^2`
Find the values of c so that for all real x, the vectors `"xc"hat"i" - 6hat"j" + 3hat"k"` and `"x"hat"i" + 2hat"j" + 2"cx"hat"k"` make an obtuse angle.
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.
Suppose that all sides of a quadrilateral are equal in length and opposite sides are parallel. Use vector methods to show that the diagonals are perpendicular.
If a line makes angles 90°, 135°, 45° with the X-, Y- and Z-axes respectively, then find its direction cosines.
If `bar"a" = 2hat"i" + 3hat"j" - hat"k"`, `bar"b" = hat"i" - 4hat"j" + 2hat"k"`, find `(bar"a" + bar"b") xx (bar"a" - bar"b")`
Find `bar"u".bar"v"` if `|bar"u"| = 2, |bar"v"| = 5, |bar"u" xx bar"v"| = 8`
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"`.
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
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")`
If the line r = `(hat"i" - 2hat"j" + 3hat"k") + lambda(2hat"i" + hat"j" + 2hat"k")` is parallel to the plane `"r" * (3hat"i" - 2hat"j" + "m"hat"k")` = 10, then the value of m is ______.
If `overlinea = hati + hatj + hatk` and `overlinec = hatj - hatk` and `overlineb` is a vector satisfying `overlinea xx overlineb = overlinec` and `overlinea . overlineb = 3`, then `3|overlineb|^2` is equal to ______
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.
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 ______.
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`
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 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`
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`
