Question

find the dot and croos product of the given vector. a= 3i-4j+ 5k and b= 6i+ 4j+ 2k​

Answer 1

$\vec{a}. \vec{b} = 12 \: \: \: and \: \: \: \vec{a} \times \vec{b} = - 28i + 24j + 36k$

Explanation:

$Given, \: \: \: \vec{a} = 3i - 4j + 5k \: \: \: and \: \: \: \vec{b} = 6i + 4j + 2k \\ we \: \: know \: \: \boxed{ \vec{a}. \vec{b} = a_{x}b_{x} + a_{y}b_{y} + a_{z}b_{z} } \\ Now, \: \: \: \vec{a}. \vec{b} = (3i - 4j + 5k).(6i + 4j + 2k) \\ \: \: \: \: \: \: = 3 \times 6 - 4 \times 4 + 5 \times 2 \\ = 18 - 16 + 10 \\ = \boxed{12} \\ we \: \: know \: \: \boxed{ \vec{a} \times \vec{b} = (a_{y}b_{z} - a_{z}b_{y})i - (a_{x}b_{z} - a_{z}b_{x})j + (a_{x}b_{y} -a_{y}b_{x} )k} \\ Now, \: \: \: \vec{a} \times \vec{b} = (3i - 4j + 5k) \times (6i + 4j + 2k) \\ = ( - 4 \times 2 - 5 \times 4) i- (3 \times 2 - 5 \times 6)j + (3 \times 4 - ( - 4 )\times 6)k \\ = ( - 8 - 20)i - (6 - 30)j + (12 + 24)k \\ = \boxed{- 28i + 24j + 36k}$