Advertisements
Advertisements
Question
Prove by vector method that the parallelograms on the same base and between the same parallels are equal in area
Solution
`bar"AB" = bar"a"`
`bar"AD" = bar"b"`
Vector area of the parallelogram is `bar"b" xx bar"a"` ........(1)
Consider the parallelogram ABB’A’
`bar"AB" = bar"a", bar"AB" = "m"bar"a"`
Because `bar"A'B"` is parallel to `bar"AB"`
Consider the triangle ADA’
By law of vectors AA’ = `"m"bar"a" + bar"b"`
Hence the vector area of parallelogram
ABB'A = `bar"a" xx ("m"bar"a" + bar"b")`
= `"m"(bar"a" xx bar"a") + (bar"a" xx bar"b")`
= 0 + `(bar"a" xx bar"b")`
= `bar"a" xx bar"b"` .......(2)
Hence the vector area of parallelogram
By (1) and (2)
Area of ABCD = Area of ABB’A’
Hence proved.
APPEARS IN
RELATED QUESTIONS
Prove by vector method that if a line is drawn from the centre of a circle to the midpoint of a chord, then the line is perpendicular to the chord
Prove by vector method that the median to the base of an isosceles triangle is perpendicular to the base
Prove by vector method that an angle in a semi-circle is a right angle
Prove by vector method that the diagonals of a rhombus bisect each other at right angles
If G is the centroid of a ΔABC, prove that (area of ΔGAB) = (area of ΔGBC) = (area of ΔGCA) = `1/3` (area of ΔABC)
Prove by vector method that sin(α + ß) = sin α cos ß + cos α sin ß
Find the magnitude and direction cosines of the torque of a force represented by `3hat"i" + 4hat"j" - 5hat"k"` about the point with position vector `2hat"i" - 3hat"j" + 4hat"k"` acting through a point whose position vector is `4hat"i" + 2hat"j" - 3hat"k"`
Choose the correct alternative:
If `vec"a"` and `vec"b"` are parallel vectors, then `[vec"a", vec"c", vec"b"]` is equal to
Choose the correct alternative:
If a vector `vecalpha` lies in the plane of `vecbeta` and `vecϒ`, then
Choose the correct alternative:
If `vec"a"*vec"b" = vec"b"*vec"c" = vec"c"*vec"a"` = 0, then the value of `[vec"a", vec"b", vec"c"]` is
Let A, B, C be three points whose position vectors respectively are
`vec"a" = hat"i" + 4hat"j" + 3hat"k"`
`vec"b" = 2hat"i" + αhat"j" + 4hat"k", α ∈ "R"`
`vec"c" = 3hat"i" - 2hat"j" + 5hat"k"`
If α is the smallest positive integer for which `vec"a", vec"b", vec"c"` are noncollinear, then the length of the median, in ΔABC, through A is ______.
Let `veca = hati - 2hatj + 3hatk, vecb = hati + hatj + hatk` and `vecc` be a vector such that `veca + (vecb xx vecc) = vec0` and `vecb.vecc` = 5. Then, the value of `3(vecc.veca)` is equal to ______.
Let `veca, vecb` and `vecc` be three unit vectors such that `|veca - vecb|^2 + |veca - vecc|^2` = 8. Then find the value of `|veca + 2vecb|^2 + |veca + 2vecc|^2`
The vector `vecp` perpendicular to the vectors `veca = 2hati + 3hatj - hatk` and `vecb = hati - 2hatj + 3hatk` and satisfying the condition `vecp.(2hati - hatj + hatk)` = –6 is ______.
Let a, b, c ∈ R be such that a2 + b2 + c2 = 1. If `a cos θ = b cos(θ + (2π)/3) = c cos(θ + (4π)/3)`, where θ = `π/9`, then the angle between the vectors `ahati + bhatj + chatk` and `bhati + chatj + ahatk` is ______.
If `vecx` and `vecy` be two non-zero vectors such that `|vecx + vecy| = |vecx|` and `2vecx + λvecy` is perpendicular to `vecy`, then the value of λ is ______.
