Given vectors are
\(\vec u=3\hat i+2\hat j-4\hat k,\)
\(\vec v=0\hat i+2\hat j+\hat k\)
and \(\vec w=10\hat i-3\hat j+6\hat k\)
Now, \(\vec u.\vec v=(3\hat i+2\hat j-4\hat k).(0\hat i+2\hat j+\hat k)\)
= 3 x 0 + 2 x 2 - 4 x 1 (∵ \(\hat i.\hat i=\hat j.\hat j=\hat k.\hat k=1\) And \(\hat i.\hat j=\hat j.\hat k=\hat k.\hat i=0)\)
= 4 - 4 = 0
∴ \(\vec u\,and\,\vec v\) are perpendicular vectors.
And \(\vec v.\vec w=(0\hat i+2\hat j+\hat k).(10\hat i-3\hat j+6\hat k)\)
= 0 x 10 + 2 x -3 + 1 x 6
= -6+6 = 0
∴ \(\vec v\,and\,\vec w\) are perpendicular vectors.
Also \(\vec u.\vec w=(3\hat i+2\hat j-4\hat k).(10\hat i-3\hat j+6\hat k)\)
= 3 x 10 + 2 x -3 - 4 x 6
= 30 - 6 - 24
= 30 - 30 = 0
∴ \(\vec u\) and \(\vec w\) are perpendicular vectors.
Therefore, \(\vec u.\vec v\) and \(\vec w\) are mutually perpendicular vectors.
