Question

7.Find the angle between the vectors bar(i)+2bar(j)+3bar(k) and 3bar(i)-bar(j)+2bar(k)

Answer 1

SOLUTION

TO DETERMINE

The angle between the vectors

$\hat{ \imath} + 2 \hat{ \jmath} + 3 \hat{k} \: \: and \: \: \: 3\hat{ \imath} - \hat{ \jmath} + 2 \hat{k}$

EVALUATION

Here the given two vectors are

$\hat{ \imath} + 2 \hat{ \jmath} + 3 \hat{k} \: \: and \: \: \: 3\hat{ \imath} - \hat{ \jmath} + 2 \hat{k}$

Let θ be the required angle

$\displaystyle \sf{ \cos \theta = \frac{( \hat{ \imath} + 2 \hat{ \jmath} + 3 \hat{k}).( 3\hat{ \imath} - \hat{ \jmath} + 2 \hat{k})}{ |( \hat{ \imath} + 2 \hat{ \jmath} + 3 \hat{k}) || ( 3\hat{ \imath} - \hat{ \jmath} + 2 \hat{k})| } }$

$\displaystyle \sf{ \implies \cos \theta = \frac{3 - 2 + 6}{ \sqrt{14} \times \sqrt{14} } }$

$\displaystyle \sf{ \implies \cos \theta = \frac{7}{ 14} }$

$\displaystyle \sf{ \implies \cos \theta = \frac{1}{ 2} }$

$\displaystyle \sf{ \implies \theta = \frac{\pi}{ 3} }$

So the required angle

$\displaystyle \sf{ = \frac{\pi}{ 3} }$

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