# If aijka¯=i^-2j^+3k^ , bijkb¯=4i^-3j^+k^ , cijkc¯=i^-j^+2k^ verify that abcabaca¯×(b¯+c¯)=a¯×b¯+a¯×c¯ - Mathematics and Statistics

Sum

If bar"a" = hat"i" - 2hat"j" + 3hat"k"  , bar"b" = 4hat"i" - 3hat"j" + hat"k" , bar"c" = hat"i" - hat"j" + 2hat"k" verify that bar"a"xx(bar"b" + bar"c") = bar"a" xx bar"b" + bar"a" xx bar"c"

#### Solution

Given: bar"a" = hat"i" - 2hat"j" + 3hat"k"  , bar"b" = 4hat"i" - 3hat"j" + hat"k" , bar"c" = hat"i" - hat"j" + 2hat"k"

∴ bar"b" + bar"c" = (4hat"i" - 3hat"j" + hat"k") + (hat"i" - hat"j" + 2hat"k")

= 5hat"i" - 4hat"j" + 3hat"k"

and bar"a" xx (bar"b" + bar"c") = |(hat"i",hat"j",hat"k"),(1,-2,3),(4,-3,1)|

= (- 6 + 12)hat"i" - (3 - 15)hat"j" + (- 4 +10)hat"k"

= 6hat"i" + 12hat"j" + 6hat"k"    ...(1)

Also, bar"a" xx bar"b" = |(hat"i",hat"j",hat"k"),(1,-2,3),(4,-3,1)|

= (- 2 + 9)hat"i" - (1 - 12)hat"j" + (- 3 + 8)hat"k"

= 7hat"i" + 11hat"j" + 5hat"k"

and bar"a" xx bar"c" = |(hat"i",hat"j",hat"k"),(1,-2,3),(1,-1,2)|

= (- 4+3)hat"i" - (2 - 3)hat"j" + (- 1 + 2)hat"k"

= - hat"i" + hat"j" + hat"k"

∴ bar"a" xx bar"b" + bar"a" xx bar"c" = (7hat"i" + 11hat"j" + 5hat"k") + (- hat"i" + hat"j" + hat"k")

= 6hat"i" + 12hat"j" + 6hat"k"     .....(2)

From (1) and (2), we get

bar"a"xx(bar"b" + bar"c") = bar"a" xx bar"b" + bar"a" xx bar"c"

Concept: Vector Product of Vectors (Cross)
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