Find a unit vector (x) with positive first coordinate orthogonal to both vector u = <8,2,-5> and vector v = <-10,-3,-5>

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tiburtius | High School Teacher | (Level 2) Educator

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Two vectors are orthogonal if their scalar product is equal to 0. So let's first find vector `y=<<y_1,y_2,y_3>>` which is orthogonal to `u`  and `v`. To do that we will solve the following system of equations:

`yu=8y_1+2y_2-5y_3=0`

`yv=-10y_1-3y_2-5y_3=0`

We have two equations with three unknowns, hence infinitely many solutions. Let's choose the solution with `y_1=1`.

`8+2y_2-5y_3=0 `                 (1)

`-10-3y_2-5y_3=0`               (2)

(1)-(2)= `18+5y_2=0 => y_2=-18/5`

`8+2(-18/5)-5y_3=0 => y_3=4/25`

Hence `y=<<1,-18/5,4/25>>`. To get unit vector `x` we need to devide `y` by its length `||y||=sqrt(y_1^2+y_2^2+y_3^2)`

`||y||=sqrt(1+324/25+16/625)=sqrt(8741/625)=sqrt(8741)/25`

`x=sqrt(8741)/25<<1,-18/5,4/25>>`

`x=<<sqrt(8741)/25,-(18sqrt(8741))/125,(4sqrt(8741))/625>>`

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