Vector product - Area
Last updated at April 16, 2024 by Teachoo
Misc 10 The two adjacent sides of a parallelogram are 2đ Ě â 4đ Ě + 5đ Ě and đ Ě â 2đ Ě â 3đ Ě Find the unit vector parallel to its diagonal. Also, find its area. Let đ â and đ â are adjacent side of a parallelogram, where đ â = 2đ Ě â 4đ Ě + 5đ Ě đ â = đ Ě â 2đ Ě â 3đ Ě Let diagonal be đ â Hence, đ â = đ â + đ â = (2đ Ě â 4đ Ě + 5đ Ě) + (1đ Ě â 2đ Ě â 3đ Ě) = (2 + 1) đ Ě â (4 + 2) đ Ě + (5 â 3)đ Ě = 3đ Ě â 6đ Ě + 2đ Ě Magnitude of âđ â â = â((3)^2+(â6)^2+(2)^2 ) = â(9+36+4) = â49 = 7 Unit vector in direction of đ â = 1/(đđđđđđĄđ˘đđ đđ đ â ) Ă đ â đ Ě = đ/đ ("3" đ Ě" â 6" đ Ě" + 2" đ Ě ) Finding Area of parallelogram Area of parallelogram = |đ â Ă đ â | Now, đ â Ă đ â = |â 8(đ Ě&đ Ě&đ Ě@2&â4&5@1&â2&â3)| = đ Ě (12 â (â10) â đ Ě (â6 â5) + đ Ě (â4 â (â4)) = 22đ Ě + 11đ Ě + 0đ Ě = 22đ Ě + 11đ Ě So đ â Ă đ â = 22đ Ě + 11đ Ě Now, |đ â Ă đ â | = â(22^2+11^2 ) = â(2^2 (ă11)ă^2+11^2 ) = â( 11^2 (2^2+1)) = 11â5 Hence, Area of parallelogram = |đ â Ă đ â |= đđâđ