# 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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### 1 Answer

You need to evaluate the cross product of the vectors `bar u` and `bar v` , such that:

`bar u x bar v = |bar u|*|bar v|*(sin theta)*bar n`

`bar n` represents the unit vector that is perpendicular to the plane

Since `sin theta = 90^o` yields:

`bar u x bar v = |bar u|*|bar v|*bar n`

You need to evaluate the lengths `|bar u|` and `|bar v|` , such that:

`|bar u| = sqrt(8^2 + 2^2 + (-5)^2) => |bar u| = sqrt 93`

`|bar v| = sqrt((-10)^2 + (-3)^2 + (-5)^2) => |bar v| = sqrt 134`

`|bar u|*|bar v| = sqrt 12462 => |bar u|*|bar v| = 111.63`

`bar u x bar v = 111.63*(bar i + bar j + bar k)`

**Hence, evaluating the unit vector orthogonal to both ` bar u` and `bar v` , using the cross product, yields **`bar u x bar v = 111.63*(bar i + bar j + bar k).`