Question

If vec A=-2hat imath+3hat imath-4hat k and vec B=3hat imath-3hat j math-5hat k find vec A*vec B​

Answer 1

SOLUTION

GIVEN

$\vec{A} = - 2 \hat{ \imath} + 3 \hat{ \jmath} - 4\hat{ k}$

$\vec{B} = 3 \hat{ \imath} - 3 \hat{ \jmath} - 5\hat{ k}$

TO DETERMINE

$\vec{A} .\vec{B}$

FORMULA TO BE IMPLEMENTED

In Vector Dot product

$(\hat{ \imath}.\hat{ \imath}) = (\hat{ \jmath}.\hat{ \jmath}) = (\hat{ k}.\hat{ k}) = 1$

$(\hat{ \imath}.\hat{ \jmath}) = (\hat{ \jmath}.\hat{ \imath}) = 0$

$(\hat{ \imath}.\hat{k}) = (\hat{k}.\hat{ \imath}) = 0$

$(\hat{ \jmath}.\hat{ k}) = (\hat{k}.\hat{ \jmath}) = 0$

EVALUATION

Here it is given that

$\vec{A} = - 2 \hat{ \imath} + 3 \hat{ \jmath} - 4\hat{ k}$

$\vec{B} = 3 \hat{ \imath} - 3 \hat{ \jmath} - 5\hat{ k}$

Now

$\vec{A} .\vec{B}$

$= ( - 2 \hat{ \imath} + 3 \hat{ \jmath} - 4\hat{ k}).( 3 \hat{ \imath} - 3 \hat{ \jmath} - 5\hat{ k})$

$= - 6(\hat{ \imath}.\hat{ \imath}) + 6(\hat{ \imath}.\hat{ \jmath}) + 10(\hat{ \imath}.\hat{k}) + 9(\hat{ \jmath}.\hat{ \imath}) - 9 (\hat{ \jmath}.\hat{ \jmath}) - 15(\hat{ \jmath}.\hat{ k}) - 12(\hat{k}.\hat{ \imath}) + 12(\hat{k}.\hat{ \jmath}) + 20(\hat{ k}.\hat{ k})$

$= - 6 - 9 + 20$

$= 5$

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