We have \(\vec A=3\hat i-2\hat j+5\hat k\) and \(\vec B=6\hat i-7\hat j+4\hat k\)
(a) A2 = \(\vec A.\vec A\) = \((3\hat i-2\hat j+5\hat k).(3\hat i-2\hat j+5\hat k)\)
= 3 x 3 + (-2) x (-2) + 5 x 5
= 9 + 4 + 25 = 38
(b) B2 = \(\vec B.\vec B\) = \((6\hat i-7\hat j+4\hat k).(6\hat i-7\hat j+4\hat k)\)
= 6 x 6 + (-7) x (-7) + 4 x 4
= 36 + 49 + 16 = 101.
(c) \(\vec A.\vec B\) = \((3\hat i-2\hat j+5\hat k).(6\hat i-7\hat j+4\hat k)\)
= 3 x 6 - 2 x -7 + 5 x 4
= 18 + 14 + 20 = 52
\((\vec A.\vec B)^2\) = 522 = 2704
(d)
\(\vec A\times\vec B=\begin{vmatrix}\hat i&\hat j&\hat k\\3&-2&5\\6&-7&4\end{vmatrix}\)
= \(\hat i\)(-2 x 4 - (-7) x 5) - \(\hat j\)(3 x 4 - 6 x 5) + \(\hat k\) (3 x -7 - 6 x -2)
= \(\hat i\) (-8 + 35) - \(\hat j\) (12 - 30) + \(\hat k\) (-21 + 12)
= 27\(\hat i\) + 18\(\hat j\) - 9 \(\hat k\)
(e) \((\vec A\times\vec B).\vec B\) = (27 \(\hat i\) + 18 \(\hat j\) - 9 \(\hat k\)). (6 \(\hat i\) - 7 \(\hat j\) + 4 \(\hat k\))
= 27 x 6 +18 x -7 - 9 x 4
= 162 -126 -36
= -2
\(((\vec A\times\vec B))^2\) = (-2)2 = 4