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Engineering
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Three vectors \vec{a} ,\vec{b} ,and \: \vec{c} are such that  \left | \vec{a} \right |=1 ,\left | \vec{b} \right |=2 , \: \left | \vec{c} \right |=4 and \vec{a}+\vec{b}+\vec{c}=\vec{0}

 Then the value of    4\; \vec{a}\cdot \vec{b}+3\; \vec{b}\cdot \; \vec{c}+3\; \vec{c}\cdot \; \vec{a} is equal to :

  • Option 1)

    27

  • Option 2)

    -68

  • Option 3)

    -26

  • Option 4)

    - 34

 
As we learnt in  Scalar Product of two vectors (dot product) - - wherein  is the angle between the vectors     Squaring  Option 1) 27 This option is incorrect  Option 2) -68 This option is incorrect  Option 3) -26 This option is correct  Option 4) - 34 This option is incorrect 
Engineering
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The perpendicular distance from the point (3, 1, 1) on the plane passing through the point (1, 2, 3) and containing the line,
\vec{r}=\hat{i}+\hat{j}+\lambda (2\hat{i}+\hat{j}+4\hat{k}),is
  • Option 1)

    \frac{3}{\sqrt{11}}

  • Option 2)

    \frac{1}{\sqrt{11}}

  • Option 3)

    \frac{4}{\sqrt{41}}

  • Option 4)

    0

 
As we learnt in  Unit vector perpendicular to the plane of vector a and b - - wherein Here  and  are two vectors.   Plane passing through (1,2,3) Containing line Normal vector of plane  Equation of plane x+6y-2z=1+12-6 x+6y-2z=7 Distance =     Option 1) This option is incorrect. Option 2) This option is incorrect. Option 3) This option is incorrect. Option 4) This option is correct.
Engineering
112 Views   |  

 Let  \underset{a}{\rightarrow} =\: 2\: \hat{i}\: +\: \hat{j}\: -2\: \hat{k} and\underset{b}{\rightarrow}= \hat{i}\: +\hat{j}.

Let \underset{c}{\rightarrow}be a vector such that

 and the angle between\underset{c}{\rightarrow}and

thenis equal to:

  • Option 1)

    2

  • Option 2)

    5

  • Option 3)

    \frac{1}{8}

  • Option 4)

    \frac{25}{8}

 
As we learnt in Vector Triple Product - - wherein x and y are scalars.   and   Scalar Product of two vectors - - wherein   is the angle between the vectors    Also,   Also,  So,      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
115 Views   |  

The area (in sq. units) of the parallelogram whose diagonals are along the vectors

\small 8 \hat{i}- 6 \hat{j} \; \; and\: \: 3 \hat{i} + 4\hat{j} -12\hat{k},

  • Option 1)

    26

  • Option 2)

    65

  • Option 3)

    20

  • Option 4)

    52

 
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    Position vector of a point - - wherein    Area of parallelogram =      where  are diagonals magnitude     Option 1) 26 This option is incorrect. Option 2) 65 This option is correct. Option 3) 20 This...
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