Calculus - Components of a unit vector

Question

Determine the components of a unit vector perpendicular to (0, 3, -5) and (2, 3, 1). I think I should be using either cross or dot product, but am unsure on what to do from there.

Answer 1

First calculate the determinant $$ \left| \begin{array}{ccc} i & j & k \\ 0 & 3& -5 \\ 2 & 3 & 1 \end{array} \right| $$ You get a vector perpendicular to both given vectors then normalize it by dividing the components by the length of it.

Edit: See this link from wikipedia.

Answer 2

"In mathematics, the cross product or vector product is a binary operation on two vectors in three-dimensional space. It results in a vector which is perpendicular to both and therefore normal to the plane containing them."

http://en.wikipedia.org/wiki/Cross_product

(furthermore, once you have the vector, you need to unitize it - http://en.wikipedia.org/wiki/Unit_vector)

Answer 3

Using the dot product: let $u=(a,b,c)\in\mathbb{R}^3$ be such a vector, and write $x=(0,3,-5)$, $y=(2,3,1)$. Then, the orthogonality conditions mean that $$ \begin{align*} 0 &= \langle u,x\rangle = 0\cdot a + 3\cdot b+(-5)\cdot c = 3b-5c \\ 0 &= \langle u,y\rangle = 2\cdot a + 3\cdot b+1\cdot c = 2a+3b+c \end{align*} $$ and the unit vector condition further imposes $$ \begin{align*} 1 &= a^2+b^2+c^2 \end{align*} $$ You thus have 3 constraints (two of them linear), and 3 unknowns. This will be enough to find solutions for $u$ (there is no unicity, as if $u$ is solution, so is $-u$, for instance).