Given `a=1/7(2 hat i+3 hat j+6 hat k),vec b=1/7(3 hat i-6 hat j+2 hat k),vec c=1/7(6 hat i+2 hat j-3 hat k), hat i, hat j,hat k` being a right handed orthogonal system of unit vectors in space, show that `vec a, vec b` and `vec c` is also another system.

Explanation

Given
`vec{a}=frac{1}{7}(2 hat{i}+3 hat{j}+6 hat{k})`

`vec{b}=frac{1}{7}(3 hat{i}-6 hat{j}+2 hat{k})`

`vec{c}=frac{1}{7}(6 hat{i}+2 hat{j}-3 hat{k})`

`vec{a} times vec{b}=(frac{1}{7})(frac{1}{7})|hat{i} \ hat{j} \hat{k} \ 2 \ 3 \ 6\ 3 \ -6 \2|`

`=frac{1}{49}(42 hat{i}+14 hat{j}-21 hat{k})`

`=frac{1}{49}[7(6 hat{i}+2 hat{j}-3 hat{k})]`

`=frac{1}{7}(6 hat{i}+2 hat{j}-3 hat{k})`

`=vec{c}`

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