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Three Dimension...

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  • Question 1
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    If the direction cosines of two lines are given by $$l+m+n=0$$ and $$l^2-5m^2+n^2=0$$, then the angle between them is

  • Question 2
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    Directions For Questions

    Let $$A$$ be the given point whose position vectors with reference to origin $$O$$ be $$ \overrightarrow{a}$$ and $$ \overrightarrow{ON}= \overrightarrow{n}$$ Let $$P$$ be any point such that $$\overline{OP}= \overrightarrow{r}$$ lies on the plane & passes through $$A$$ and orthogonal to $$ON$$. Then for any point $$P$$ lies on the plane then.$$\overrightarrow{AP} \perp  \overrightarrow{n}$$
    $$\displaystyle \therefore \overrightarrow{AP}.\ \overrightarrow{n}=0$$
    $$\displaystyle \Rightarrow \left ( \overrightarrow{OP}-\overrightarrow{OA} \right ).\overrightarrow{n}=0$$
    $$\displaystyle \Rightarrow  \overrightarrow{r}\cdot \overrightarrow{n}=\overrightarrow{a}.\overrightarrow{n}$$       $$($$Knows as scalar product form$$)$$
    $$\displaystyle \Rightarrow  \overrightarrow{r}\cdot \overrightarrow{n}=p,$$ where $$p$$ is the $$\perp$$er distance from origin to the plane.
    On the bases of above information answer the following questions

    ...view full instructions

    The, position vector of the foot of the $$\perp$$er drawn from origin to the plane is $$\displaystyle  4\hat{i}-2\hat{j}-5\hat{k} $$ then equation of the plane is

  • Question 3
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    Directions For Questions

    Consider the lines:

     $$\displaystyle L_{1}=\dfrac{x+1}{3}=\dfrac{y+2}{1}=\dfrac{z+1}{2}$$ and  $$\displaystyle L_{2}=\dfrac{x-2}{1}=\dfrac{y+2}{2}=\dfrac{z-3}{3}$$

    On the basis of the above information, answer the following question:

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    The vector equation of the line $$\displaystyle L_{1}$$ is $$\displaystyle a+\lambda \overline{b}$$ then $$\displaystyle \overline{a}$$ equals

  • Question 4
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    If $$A$$ , $$B$$ and $$C$$ are three collinear points, where $$A= i + 8 j - 5k $$, $$ B  = 6i-2j$$ and $$C= 9i + 4j - 3 k$$, then $$B$$ divides $$AC$$ in the ratio of :

  • Question 5
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    If the points $$a(cos \alpha + i sin \alpha)$$ , $$b(cos \beta + i sin \beta)$$ and $$c(cos \gamma + isin \gamma)$$ are collinear then the value of $$|z|$$ is:  
    ( where $${z = bc  \ sin(\beta-\gamma) + ca \ sin(\gamma-\alpha) + ab \ sin(\alpha - \beta) + 3i -4k}$$ )

  • Question 6
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    The angle between two diagonals of a cube is

  • Question 7
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    Equation of the plane through the mid-point of the line segment joining the points $$P(4, 5, -10), \,Q(-1, 2, 1)$$ and perpendicular to $$PQ$$ is

  • Question 8
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    If $$(2, -1, 3)$$ is the foot of the perpendicular drawn from the origin to the plane, then the equation of the plane is

  • Question 9
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    What is the angle between the lines $$\cfrac { x-2 }{ 1 } =\cfrac { y+1 }{ -2 } =\cfrac { z+2 }{ 1 } $$ and $$\cfrac { x-1 }{ 1 } =\cfrac { 2y+3 }{ 3 } =\cfrac { z+5 }{ 2 } =?$$

  • Question 10
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    A plane mirror is placed at the origin so that the direction ratios of its normal are $$(1,-1,1)$$. A ray of light, coming along the positive direction of the x-axis, strikes the mirror. The direction cosines of the reflected ray are

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