If p, q and r are perpendicular to q + r, r + p and p + q respectively and if |p + q| = 6, |q + r| = `4sqrt(3)` and |r + p| = 4, then |p + q + r| is

Options

A. `5sqrt(2)`
B. 10
C. 15
D. 5

Answer

A

Explanation

`|p+q|=6implies |p+q|^(2)=36`
`implies p^(2) +q^(2)+2p.q=36`
Similarly `q^(2)+r^(2)+2q.r=48`
` and r^(2) +p^(2)+2r .p =16`
Adding all we get
`2( p^(2)+q^(2)r^(2)+p.q+q.r+r.p)=36+48+16`
`implies 2(p^(2) +q^(2)+r^(2))=100`
`[ :' p.q+q.r+r.p=0]`
`implies p^(2)+q^(2)+r^(2)=50`
`=|p+q+r|^(2)=50`
`implies |p+q+r|=5sqrt(2)`