If `bar(a),bar(b),bar(c)` are non-coplanar unit vectors each including the angle of measure `30^(@)` with the other, then find the volume of tetrahedron whose co-terminal edges are `bar(a),bar(b),bar(c)`.

Explanation

`bar(a),bar(b)bar(c)` are three non - coplanar unit vectors each including the angle of measure `30^(@)`.
`:.bar(a)*bar(a)=bar(b)*bar(b)=bar(c)*bar(c)=1`
`bar(a)*bar(b)=bar(b)*bar(a)=1xx1xxcos30^(@)=(sqrt(3))/(2)`
`bar(a)*bar(c)=bar(c)*bar(a)=1xx1xxcos30^(@)=(sqrt(3))/(2)`
`bar(b)*bar(c)=bar(c)*bar(b)=1xx1xxcos30^(@)=(sqrt(3))/(2)`
Now, volume of tetrahedron `=(1)/(6)[bar(a)bar(b)bar(c)]`
`:.("volume")^(2)=(1)/(36)([bar(a)bar(b)bar(c)])^(2)`
`=(1)/(36)|{:(bar(a)*bar(a)" "bar(a)*bar(b)" "bar(a)*bar(c)),(bar(b)*bar(a)" "bar(b)*bar(b)" "bar(b)*bar(c)),(bar(c)*bar(a)" "bar(c)*bar(b)" "bar(c)*bar(c)):}|`
`=(1)/(36)|{:(1,(sqrt(3))/(2),(sqrt(3))/(2)),((sqrt(3))/(2),1,(sqrt(3))/(2)),((sqrt(3))/(2),(sqrt(3))/(2),1):}|`
`=(1)/(36)[1(1-(3)/(4))-(sqrt(3))/(2)((sqrt(3))/(2)-(3)/(4))+(sqrt(3))/(2)((3)/(4)-(sqrt(3))/(2))]`
`=(1)/(36)((3sqrt(3))/(4)-(5)/(4))=(3sqrt(3)-5)/(144)`
`:.` volume of tetrahedron `=(sqrt(3sqrt(3)-5))/(12)` cu units