Mathematics Area Of Triangle and Parrallelogram

Please Wait... While Loading Full Video

Area Of Triangle and Parrallelogram

Area of a triangle :

1. Area of triangle `ABC =1/2 ab sin theta =1/2 |vec(a)xxvec(b) | =1/2 |vec(AB)xxvec(AC) |`

2. If `vec(a) ,vec(b),vec(c)` are position vectors of vertices of a `Delta ABC` then its

Area `= 1/2 |(vec(a)xxvec(b)) +(vec(b)xxvec(c)) +(vec(c)*vec(a)) |`
Note:

`(i)` lf 3 points with position vectors `vec(a) ,vec(b)` and `vec(c)` are collinear then `vec(a) xx vec(b) + vec(b) xx vec(c)+ vec(c) xx vec(a)=0`

`(ii)` Unit vector perpendicular to the plane of the `Delta ABC` when `vec(a),vec(b),vec(c)` are the p.v. of its angular point is

` hat(n) = pm ( vec(a) xx vec(b) + vec(b) xx vec(c) + vec(c) xx vec(a) )/(2Delta)` ,where `vec(a),vec(b),vec(c)` are the position vectors of the angular points of the triangle `ABC` .

1. Area of triangle `ABC =1/2 ab sin theta =1/2 |vec(a)xxvec(b) | =1/2 |vec(AB)xxvec(AC) |`

2. If `vec(a) ,vec(b),vec(c)` are position vectors of vertices of a `Delta ABC` then its

Area `= 1/2 |(vec(a)xxvec(b)) +(vec(b)xxvec(c)) +(vec(c)*vec(a)) |`
Note:

`(i)` lf 3 points with position vectors `vec(a) ,vec(b)` and `vec(c)` are collinear then `vec(a) xx vec(b) + vec(b) xx vec(c)+ vec(c) xx vec(a)=0`

`(ii)` Unit vector perpendicular to the plane of the `Delta ABC` when `vec(a),vec(b),vec(c)` are the p.v. of its angular point is

` hat(n) = pm ( vec(a) xx vec(b) + vec(b) xx vec(c) + vec(c) xx vec(a) )/(2Delta)` ,where `vec(a),vec(b),vec(c)` are the position vectors of the angular points of the triangle `ABC` .

Vector Area of a parallelogram and a quadrilateral `ABCD`

The vector product of the vectors `vec(a)` and `vec(b)` represents a vector whose modulus is equal to the area of the parallelogram whose two adjacent sides are represented by `vec(a)` and `vec(b)`

Area of parallelogram ` = base quad xx height = ab sin theta = |vec(a) xx vec(b) |`

Area of quadraliteral if its diagonal vectors are `vec(d_1)` & `vec(d_2)` is given by` = 1/2 |vec(d_1)xxvec(d_2) |`

Vector Area of a quadrilateral `ABCD` = Vector area of `Delta ABC+` vector area of `Delta ACD`

`= 1/2 (vec(AB) xx vec(AC)) +1/2 (vec(AC) xx vec(AD))`

`=1/2 ( vec(AB) xx vec(AC) - vec(AD) xx vec(AC)) =1/2 (vec(AB) - vec(AD)) xx vec(AC)`

`= 1/2 (vec(AB) + vec(DA))xx vec(AC) =1/2 vec(DB) xx vec(AC)`

`:.` Area of square `ABCD =1/2 |vec(DB) xx vec(AC) | =1/2 |vec(AC) xx vec(BD) |`

The vector product of the vectors `vec(a)` and `vec(b)` represents a vector whose modulus is equal to the area of the parallelogram whose two adjacent sides are represented by `vec(a)` and `vec(b)`

Area of parallelogram ` = base quad xx height = ab sin theta = |vec(a) xx vec(b) |`

Area of quadraliteral if its diagonal vectors are `vec(d_1)` & `vec(d_2)` is given by` = 1/2 |vec(d_1)xxvec(d_2) |`

Vector Area of a quadrilateral `ABCD` = Vector area of `Delta ABC+` vector area of `Delta ACD`

`= 1/2 (vec(AB) xx vec(AC)) +1/2 (vec(AC) xx vec(AD))`

`=1/2 ( vec(AB) xx vec(AC) - vec(AD) xx vec(AC)) =1/2 (vec(AB) - vec(AD)) xx vec(AC)`

`= 1/2 (vec(AB) + vec(DA))xx vec(AC) =1/2 vec(DB) xx vec(AC)`

`:.` Area of square `ABCD =1/2 |vec(DB) xx vec(AC) | =1/2 |vec(AC) xx vec(BD) |`

SiteLock