# For given two vectors A = 3i − 2j + 5k  and B = 6i − 7j + 4 k  find (a) A^2; (b) B^2; (c) (A · B)^2; (d)AxB; (e)((AxB).B)^2

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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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