Question

Find two unit vectors orthogonal to both ñ = (1, 2, - 4) and ñ = (0,2,6). Give exact values (no decimals). Separate the vectors with a comma. unit vectors =

Answer 1

Find a vector, lets call it c, that is perpendicular (or orthogonal) to vectors
`a= < 1,2,-4 >` and
`b= < 0,2,6 >`

`c` =
`a` x
`b` =
`\left[\begin{array}{ccc}i&j&k\\a_(x) &a_(y)&a_(z)\\b_(x)&b_(y)&b_(z)\end{array}\right]` =
`(a_(y)b_(z)-a_(z)b_(y))i-(a_(x)b_(z)-a_(z)b_(x))j+(a_(x)b_(y)-a_(y)b_(x))k`

`c` =
`\left[\begin{array}{ccc}i&j&k\\1&2&-4\\0&2&6\end{array}\right]` =
`[2(6)-(-4)(2)]i-[1(6)-(-4)(0)]j+[1(2)-2(0)]k`

=>
`(12+8)i-(6-0)j+(2-0)k`

=>
`20i-6j+2k`

Thus vector
`c= < 20,-6,2 >`, which is perpendicular (or orthogonal) to vectors
`a\\` and
`b`.