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Engineering
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Let   \overrightarrow{a},\overrightarrow{b}and\, \overrightarrow{c}   be three non-zero vectors such that no two of them are collinear and

(\overrightarrow{a}\times \overrightarrow{b})\times \overrightarrow{c}=\frac{1}{3}\left | \overrightarrow{b} \right |\, \left | \overrightarrow{c} \right |\,\overrightarrow{a}.   if  \theta is the angle between vectors \overrightarrow{b}\, and \, \overrightarrow{c},

then a value of \sin \theta  is:

  • Option 1)

    \frac{2\sqrt{2}}{3}

  • Option 2)

    \frac{-\sqrt{2}}{3}

  • Option 3)

    \frac{2}{3}

  • Option 4)

    \frac{-2\sqrt{3}}{3}

 
As we learnt in  Vector Triple Product (VTP) - - wherein are three vectors.   and   Scalar Product of two vectors (dot product) - - wherein  is the angle between the vectors     Thus     Option 1) This option is correct. Option 2) This option is incorrect. Option 3) This option is incorrect. Option 4) This option is incorrect.
Engineering
109 Views   |  

In    a    parallelogram ABCD ,   \left | \overrightarrow{AB} \right |=a,\: \left | \overrightarrow{AD} \right |=b\; and\: \left | \overrightarrow{AC} \right |=c,\; then\: \overrightarrow{DB}\cdot \overrightarrow{AB}\; has\: the\; value:

  • Option 1)

    \frac{1}{2}\left ( a^{2}-b^{2}+c^{2} \right )

  • Option 2)

    \frac{1}{4}\left ( a^{2}+b^{2}-c^{2} \right )

  • Option 3)

    \frac{1}{3}\left ( b^{2}+c^{2}-a^{2} \right )

  • Option 4)

    \frac{1}{2}\left ( a^{2}+b^{2}+c^{2} \right )

 
As we learnt in Scalar Product of two vectors (dot product) - - wherein  is the angle between the vectors Thus let A be position  vector of B is  , D is                                          ..............(i) Also here, Put in (i)     Option 1) This option is correct. Option 2) This option is incorrect. Option 3) This option is incorrect. Option 4) This option is incorrect.
Engineering
172 Views   |  

Let \vec{a}\: and \: \vec{b} be two unit vectors such that \left | \vec{a} +\vec{b}\right |=\sqrt{3}

if\vec{c}= \vec{a}+2\vec{b}+3\left ( \vec{a} \times \vec{b}\right )    then   2\left | \vec{c} \right |    is equal to:

  • Option 1)

    \sqrt{55}

  • Option 2)

    \sqrt{51}

  • Option 3)

    \sqrt{43}

  • Option 4)

    \sqrt{37}

 
As we learnt in  Vector Product of two vectors(cross product) - If  and  are two vectors and  is the angle between them , then  - wherein  is unit vector perpendicular to both    and    Scalar Product of two vectors (dot product) - - wherein  is the angle between the vectors   Squaring both sides            Option 1) This option is correct. Option 2) This option is...
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