Q. 124.4( 8 Votes )

Let , and . Find a vector which is perpendicular to both and , and .

Answer :

To Find: Find a vector d which is perpendicular to both a and b.


Explanation:





Let us Assume



If a and d are perpendicular then a.d=0



Then,


X+2y+2z=0 …(i)


If b and d are perpendicular then b.d=0



Then,


Where


3x-2y+7z=0 …(ii)


If c and d are perpendicular then c.d=15



Then,


Where


2x-y+4z=15 …(iii)


Now, We have three equations,


X+2y+2z=0 …(i)


3x-2y+7z=0 …(ii)


2x-y+4z=15 …(iii)


We can solve it by Elimination Method


Multiply by 2 in eq (i) and then subtract by (iii)


2x+4y+4z=0


2x-y+4z=15


On subtracting we get


5y=-15


y=-3


Now, put the value of y in equation (i) and (ii)


X+2(-3)+2z=0


X+2z=6 …(iv)


And,


3x-2(-3)+7z=0


2x+7z=-6 …(v)


Now, Multiply by 2 in equation (iv) and subtract by (v), then


2x+4z=12


2x+7z=-6


On subtracting we get


-3z=18


Z=6


Now, Put the value of z in equation (iv), we get


x+2(6)=6


x=6-12


x=-6


Therefore,



Where x=-6 y=-3 and z=6



Hence, This is the required vector.


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