Product of Vectors

Show that the vectors `- 2hat"i" - hat"j" - hat"k", - 3hat"i" - 4hat"j" - 4hat"k", hat"i" - 3hat"j" - 5hat"k"` form a right angled triangle

If `|vec"a"|= 5, |vec"b"| = 6, |vec"c"| = 7` and `vec"a" + vec"b" + vec"c" = vec"0"`, find `vec"a" * vec"b" + vec"b" *vec"c" + vec"c" * vec"a"`

Show that the points (2, –1, 3), (4, 3, 1) and (3, 1, 2) are collinear

If `vec"a", vec"b"` are unit vectors and q is the angle between them, show that 

`sin  theta/2 = 1/2|vec"a" - vec"b"|`

If `vec"a", vec"b"` are unit vectors and q is the angle between them, show that 

`cos  theta/2 = 1/2|vec"a" + vec"b"|`

If `vec"a", vec"b"` are unit vectors and q is the angle between them, show that 

`tan  theta/2 = |vec"a" - vec"b"|/|vec"a" + vec"b"|`

Find `vec"a"*vec"b"` when `vec"a" = hat"i" - 2hat"j" + hat"k"` and `vec"b" = 3hat"i" - 4hat"j" - 2hat"k"`

Find `vec"a"*vec"b"` when `vec"a" = 2hat"i" + 2hat"j" - hat"k"` and `vec"b" = 6hat"i" - 3hat"j" + 2hat"k"`