Advertisements
Advertisements
Question
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.
Solution
Let ABCD be a quadrilateral in which
`|bar"AB"| = |bar"BC"| = |bar"CD"| = |bar"DA"|` ....(1)
and AB || DC and AD || BC
∴ `bar"AB" = bar"DC"` and `bar"AD" = bar"BC"` ...(2)
Now, `bar"AC" = bar"AB" + bar"BC"`
and `bar"BD" = bar"BA" + bar"AD" = - bar"AB" + bar"BC"` ...[By(2)]
`= bar"BC" - bar"AB"`
∴ `bar"AC".bar"BD" = (bar"AB" + bar"BC").(bar"BC" - bar"AB")`
`= bar"AB".(bar"BC" - bar"AB") + bar"BC" . (bar"BC" - bar"AB")`
`= bar"AB".bar"BC" - bar"AB".bar"AB" + bar"BC".bar"BC" - bar"BC".bar"AB"`
`= |bar"BC"|^2 - |bar"AB"|^2` ....`[∵ bar"AB".bar"BC" = bar"BC".bar"AB"]`
= 0 ...[By(1)]
∵ `bar"AC", bar"BD"` are non-zero vectors
∴ `bar"AC"` is perpendicular to `bar"BD"`
Hence, the diagonals are perpendicular.
APPEARS IN
RELATED QUESTIONS
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.
If `hat"p", hat"q"` and `hat"r"` are unit vectors `hat"p"+hat "r" = hat "q"`, find `hat"p".hat"q".`
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`
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"`
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"`.
Find `bar"a"` if `bar"a" xx hat"i" + 2bar"a" - 5hat"j" = bar"0"`
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.
Find the direction ratios of a vector perpendicular to the two lines whose direction ratios are 1, 3, 2 and –1, 1, 2
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 ______.
The area of triangle ABC in which c = 8 , b = 3, ∠A = 60° is ______
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 `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 ______.
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`
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 `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`
