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Find a vector x perpendicular to the vectors vector u = <9,0,12> and vector v = <-4,5,-9>

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Tibor Pejić eNotes educator | Certified Educator

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Two vectors are perpendicular if their scalar product is equal to 0. Thus we get the following system of equations:

`x cdot u=9x_1+0+12x_3=0`

`x cdot v=-4x_1+5x_2-9x_3=0`

As you can see we have two equations with three unknowns hence infinitely many solutions. Since we need only 1 solution we can choose the one with for example `x_1=1`. Hence we have

`9+12x_3=0 => x_3= -3/4`

`-4+5x_2-9(-3/4)=0`

`5x_2=4-27/4`

`x_2=-11/20`

So our vector is `x=<<x_1,x_2,x_3>>=<<1,-11/20,-3/4>>`

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