Question

The vectors a b and c are such that a+b+c=0. Determine the value of a•b+a•c+b•c if |a|=1, |b|=2 and |c|=3.. here's my solution

Answer 1

Given:


`\begin{cases}\mathbf a+\mathbf b+\mathbf c=\mathbf0&(1)\\\|\mathbf a\|=1&(2)\\\|\mathbf b\|=2&(3)\\\|\mathbf c\|=3&(4)\end{cases}`

Take the dot product of both sides of (1) with
`\mathbf a`. You end up with


`\mathbf a\cdot(\mathbf a+\mathbf b+\mathbf c)=\mathbf a\cdot\mathbf0`

`\mathbf a\cdot\mathbf a+\mathbf a\cdot\mathbf b+\mathbf a\cdot\mathbf c=0`

`\|\mathbf a\|^2+\mathbf a\cdot\mathbf b+\mathbf a\cdot\mathbf c=0`

`\mathbf a\cdot\mathbf b+\mathbf a\cdot\mathbf c=-1`

Doing the same thing with
`\mathbf b` and
`\mathbf c`, you end up with the system


`\begin{cases}\mathbf a\cdot\mathbf b+\mathbf a\cdot\mathbf c=-1\\\mathbf a\cdot\mathbf b+\mathbf b\cdot\mathbf c=-4\\\mathbf a\cdot\mathbf c+\mathbf b\cdot\mathbf c=-9\end{cases}\implies \begin{cases}\mathbf a\cdot\mathbf b=2\\\mathbf a\cdot\mathbf c=-3\\\mathbf b\cdot\mathbf c=-6\end{cases}`

and so


`\mathbf a\cdot\mathbf b+\mathbf a\cdot\mathbf c+\mathbf b\cdot\mathbf c=2-3-6=-7`