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  • Answer please! - Letbe three unit vectors, out of which vectorsare non-parallel. Ifare the angles which vector - Vector Algebra - JEE Main

Let  \vec{a},\vec{b}\: \: and\: \: \vec{c} be three unit vectors, out of which vectors \vec{b}\: \: and\: \: \vec{c} are non-parallel. If \alpha \: \: and\: \: \beta are the angles which vector \vec{a} makes with vectors \vec{b}\: \: and\: \: \vec{c} respectively and \vec{a }\times \left ( \vec{b} \times \vec{c}\right )=\frac{1}{2}\: \vec{b}, then \left | \alpha -\beta \right | is equal to : 

  • Option 1)

    45^{\circ}

  • Option 2)

     

    60^{\circ}

  • Option 3)

    30^{\circ}

  • Option 4)

    90^{\circ}

Answers (1)

best_answer

 

Angle between vector a and vector b -

\cos \Theta =\frac{\vec{a}.\vec{b}}{\left | \vec{a} \right |\left | \vec{b} \right |}

- wherein

Here 0\leq \Theta \leq \pi??????

 

 

Vector Triple Product (VTP) -

\vec{a}\times \left ( \vec{b} \times \vec{c}\right )= \left ( \vec{a}.\vec{c} \right )\vec{b}-\left ( \vec{a}.\vec{b}\right )\vec{c}

\left ( \vec{a}\times \vec{b} \right )\times \vec{c}= \left ( \vec{a}.\vec{c} \right )\vec{b}-\left ( \vec{b}.\vec{c}\right )\vec{a}

- wherein

\vec{a}, \vec{b}, \vec{c}are three vectors.

 

 7

 

\Rightarrow (\vec{a}.\vec{c})\vec{b}-(\vec{a}.\vec{b})\vec{c}=\frac{\vec{b}}{2}\\\\\Rightarrow (\vec{a}.\vec{c}-\frac{1}{2})\vec{b}= (\vec{a}.\vec{c})\vec{c}\\\\\\\: \vec{b}\: is\: not\: parellel\: to\: \vec{c}\\\\\Rightarrow \vec{a}.\vec{c}=1/2\; and\: \: \vec{a}\: \vec{b}=0\\\\\Rightarrow \alpha =\pi /2\: \: and\; \beta =\pi /3\\\\\Rightarrow \left | \alpha -\beta \right |=\pi /6


Option 1)

45^{\circ}

Option 2)

 

60^{\circ}

Option 3)

30^{\circ}

Option 4)

90^{\circ}

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