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Help me answer: - Vector Algebra - JEE Main

If \vec{a},\; \vec{b},\: and\: \vec{c} are unit vectors such that \vec{a}+2 \vec{b}+2\vec{c}=\vec{0},\; \; then\: \: \left | \vec{a }\times \vec{c} \right | is equal to : 

 

  • Option 1)

    \frac{\sqrt{15}}{4}

     

     

     

  • Option 2)

    \frac{1}{4}

  • Option 3)

    \frac{15}{16}

  • Option 4)

    \frac{\sqrt{15}}{16}

 
Answers (2)
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As we learned

 

Scalar Product of two vectors (dot product) -

\vec{a}\vec{b}=\left | a \right |\left | b \right |Cos\theta

- wherein

\Theta is the angle between the vectors\vec{a}\: and\:\vec{b}

 

 

Vector Product of two vectors(cross product) -

If \vec{a} and \vec{b} are two vectors and \Theta is the angle between them , then \vec{a}\times \vec{b}=\left |\vec{a} \left | \right |\vec{b} \right |Sin\Theta \hat{n}

- wherein

\hat{n} is unit vector perpendicular to both \vec{a} \: and \: \vec{b}

 

 

Given \vec{a},\vec{b},\vec{c} are unit vectors.

2\vec{b} =-\left ( \vec{a}+2\vec{c} \right )

4=1+4+4\vec{a}\vec{c}\Rightarrow \cos \theta =-\frac{1}{4}

\tan \left | \vec{a} \times \vec{c}\right |=\left | \vec{a} \right |\left | \vec{c} \right |\sin \theta

=1\times 1\times \frac{\sqrt{15}}{4}

 

 


Option 1)

\frac{\sqrt{15}}{4}

 

 

 

Option 2)

\frac{1}{4}

Option 3)

\frac{15}{16}

Option 4)

\frac{\sqrt{15}}{16}

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