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For given two vectors A = 3i − 2j + 5k  and B = 6i − 7j + 4 k  find 

(a) A2; (b) B2; (c) (A · B)2; (d) A x B; (e) ((A x B). B)2

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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

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