Answers posted by rvidyagovindarajan_1

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If $\overrightarrow a,\overrightarrow b,\overrightarrow c$ are three coplanar vectors then, $[(2\overrightarrow a-\overrightarrow b)\:(2\overrightarrow b-\overrightarrow c)\:(2\overrightarrow c-\overrightarrow a)]=?$

answered Jan 5, 2014

Since $\overrightarrow a,\overrightarrow b,\overrightarrow c$ are coplanar, $[\overrightarrow a\:\...

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The position vector of the vertices $A,B,C$ of a $\Delta\: ABC$ are $\hat i-\hat j-3\hat k,\:2\hat i+\hat j-2\hat k\:\:and\:\:-5\hat i+2\hat j-6\hat k$ respectively. The length of the bisector $AD$ of angle $A$ where $D$ is on the line $BC$ is ?

answered Jan 5, 2014

Toolbox:Angular bisector of an angle of a $\Delta $ divides the opp. side in the ratio of the lengt...

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A line passes through the points with position vectors are $\hat i+\hat j-2\hat k$ and $\hat i-3\hat j+\hat k$. The position vector of a point on the line which is at a unit distance from the first point is ?

answered Jan 5, 2014

Toolbox:Eqn. of aline through two points $\overrightarrow a$ and $\overrightarrow b$ is $\overri...

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If $\overrightarrow a,\overrightarrow b,\overrightarrow c$ are position vectors of three non collinear points $A,B,C$ respectively, then the shortest distance of $A$ from $BC$ is ?

answered Jan 5, 2014

Toolbox:Shortest distance of a point from a line is its $\perp$ distance.Projection of $\overrighta...

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The distance of the point $(1,1,1) $ from the plane passing through the points $(2,1,1),\:(1,2,1)\:and\:(1,1,2)$ is ?

answered Jan 5, 2014

Toolbox:Distance of a point $P$ from the plane $ABC$ is length of projection of $\overrightarrow...

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The line joining the points $6\overrightarrow a-4\overrightarrow b-5\overrightarrow c$ and $-4\overrightarrow c$ and the line joining the points $-\overrightarrow a-2\overrightarrow b-3\overrightarrow c$ and $\overrightarrow a+2\overrightarrow b-5\overrightarrow c$ Intersects at the point ?

answered Jan 5, 2014

Toolbox:Eqn. of a line joining the points $\overrightarrow a\:\:and\:\:\overrightarrow b$ is $\ov...

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A unit vector coplanar with the vectors $\hat i-\hat j\:\;and\:\:\hat i+2\hat j$ and is $\perp$ to the first vector, $\hat i-\hat j$ is ?

answered Jan 5, 2014

Toolbox:If vectors $\overrightarrow a,\overrightarrow b,\overrightarrow c$ are coplanar, then $ [\o...

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If $\overrightarrow {OA}=\hat i+3\hat j-2\hat k\:\:and\:\:\overrightarrow {OB}=3\hat i+\hat j-2\hat k$ and $C$ is a point of $AB$ so that $\overrightarrow {OC}$ bisects the angle $AOB$, then $\overrightarrow {OC}=?$

answered Jan 5, 2014

Given $\overrightarrow {OA}=\hat i+3\hat j-2\hat k\:\;and\:\:\overrightarrow {OB}=3\hat i+\hat j-2\...

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The equation of the plane having intercepts of $3$ and $4$ with $x$ and $z$ axes respectively and parallel to $y$ axis is ?

answered Jan 5, 2014

Toolbox:Equation of a plane with $a,b,c$ as its respective intercepts is $\large\frac{x}{a}+\frac...

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If the lines $x=1+s,\:y=-3-\lambda s,\:z=1+\lambda s$ and $x=t/2,\:y=1+t,\:z=2-t$ with parameters $s\:and\:t$ respectively, are coplanar, then $\lambda=?$

answered Jan 4, 2014

Toolbox:The lines $\large\frac{x-x_1}{l_1}=\frac{y-y_1}{m_1}=\frac{z-z_1}{n_1}=\lambda_1$ and $\la...

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The equation of the plane containing the lines $\overrightarrow r=\overrightarrow a_1+\lambda\overrightarrow b\:\:and\:\:\overrightarrow r=\overrightarrow a_2+\mu\overrightarrow b$ is ?

answered Jan 4, 2014

Toolbox:Line joining any two points on a plane lie on the plane.Every line on the plane is $\perp$ ...

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The equation of the plane that contains the line $\overrightarrow r=2\hat i+\lambda(\hat j-\hat k)$ and $\perp$ to the plane $\overrightarrow r.(\hat i+\hat k)=3$ is ?

answered Jan 4, 2014

Let the normal to the required plane be $\overrightarrow n$ and a point on the plane be $\overr...

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The equation of the plane through the point $(1,2,-1)$ and $\perp$ to the line of intersection of the planes $\overrightarrow r.(3\hat i-\hat j+\hat k)=1$ and $\overrightarrow r.(\hat i+4\hat j-2\hat k)=2$ is?

answered Jan 4, 2014

Toolbox:Vector $\perp$ to $\overrightarrow a\:and\:\overrightarrow b\:is \:\overrightarrow a\times\...

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If the position vector of the points $A\:and\:B$ are $\hat i-\hat j+3\hat k$ and $3\hat i+3\hat j+3\hat k$ and the equation of a plane is $\overrightarrow r.(5\hat i+2\hat j-7\hat k)+9=0$, then the points $A\:and\:B$ are?

answered Jan 4, 2014

If the point satisfies the equation of the plane then it lies on the plane.If the points when substi...

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The equation of the plane through origin and the line of intersection of the planes $\overrightarrow r.\overrightarrow a=\lambda\:\:and\:\:\overrightarrow r.\overrightarrow b=\mu$ is ?

answered Jan 4, 2014

Toolbox:Equation of any plane passing through the line of intersection of the planes $P_1=0$ and $...

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If $O$ is origin, $|\overrightarrow{OP}|=3$ with $d.r.=(-1,2,-2)$ then the coordinate of $P$ is ?

answered Jan 3, 2014

Given: $d.r.$ of $\overrightarrow{OP}=(-1,2,-2)$$\Rightarrow\:P(-1,2,-2)$

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If $P(x,y,z)$ is a point on the line segment joining $Q(2,2,4)\:\:and\:\:R(3,5,6)$ such that the projections of $\overrightarrow {OP}$ on the coordinate axes are $\large\frac{13}{5},\frac{19}{5},\frac{26}{5}$ respectively, then $P$ divides $QR$ in the ratio ?

answered Jan 3, 2014

Toolbox:Projection of $\overrightarrow a$ on $\overrightarrow b$ is $\large\frac{\overrightarrow ...

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If $P$ is a point in space such that $OP=12$ and $\overrightarrow {OP} $ is inclined with angle $45^{\circ} $ with $OX$ and $60^{\circ}$ with $OY$ then the coordinates of $P$ is ?

answered Jan 3, 2014

Toolbox:If a line makes angle $\alpha,\:\beta\:and\:\gamma$ with coordinate axes, then $cos^2\alp...

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If $P(3,2,-4),\:Q(5,4,-6)\:and\:R(9,8,-10)$ are collinear,then $R$ divides $PQ$ in ratio?

answered Jan 3, 2014

Let the ratio be $\lambda:1$According to section formula, the coordinates of R is given by $(\larg...

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The length of $\perp$ from origin to the plane passing through the point $\overrightarrow a$ and containing the line $\overrightarrow r=\overrightarrow b+\lambda \overrightarrow c$ is ?

answered Jan 3, 2014

Toolbox:Vector equation of a plane through the point $\overrightarrow a$ and with normal $\overrigh...

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The equation of the plane through the line of intersection of the planes $ax+by+cz=d=0$ and $a'x+b'y+c'z+d'=0$ and parallel to the line $y=0,z=0$ is ?

answered Jan 3, 2014

Equation of any plane through the line of intersection of the given two planes$ax+by+cz+d=0$ and $...

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The line through the point $\hat i+3\hat j+2\hat k$ and $\perp$ to the lines $\overrightarrow r=(\hat i+2\hat j-\hat k)+\lambda(2\hat i+\hat j+\hat k)$ and $\overrightarrow r=(2\hat i+6\hat j+\hat k)+\mu (\hat i+2\hat j+3\hat k)$ is ?

answered Jan 3, 2014

Toolbox:$\overrightarrow a\times\overrightarrow b$ is along the direction $\perp$ to both $\ove...

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The distance of the point $(-1,2,6)$ from the line through the point $(2,3,-4)$ along the vector $6\hat i+3\hat j-4\hat k$ is ?

answered Jan 3, 2014

Equation of the line through the point $P(2,3,-4)$ along the vector $6\hat i+3\hat j-4\hat k$ is...

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The lines $\overrightarrow r=\overrightarrow a_1+\lambda\overrightarrow b_1$ and $\overrightarrow r=\overrightarrow a_2+\alpha \overrightarrow b_2$ are coplanar if ?

answered Jan 3, 2014

Toolbox:$[\overrightarrow a\:\overrightarrow b\:\overrightarrow c]=\left| \begin {array} {ccc} a_1 &...

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If the plane $\large\frac{x}{2}+\frac{y}{3}+\frac{z}{4}=1$ cuts the coordinate axis in $A,B,C$ respectively, then the area of the $\Delta\:ABC$ is ?

answered Jan 3, 2014

Toolbox:Area of $\Delta\:ABC =\large\frac{1}{2}$$|\overrightarrow {AB}\times\overrightarrow {AC}|$...

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The image of the point $P(1,3,4)$ in the plane $2x-y+z+3=0$ is ?

answered Jan 2, 2014

Let $Q$ be the image of $P(1,3,4)$.$d.r.$ of normal to the plane is $\overrightarrow n=(2,-1,1...

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If $L\rightarrow\:\:\large\frac{x-1}{3}=\frac{y+1}{2}=\frac{z-3}{-1}$ and the plane $P\rightarrow\:\:x-2y-z=0$, then which of the following is true?

answered Jan 2, 2014

For a line to be $\perp$ to a plane, $d.r.$ of the line should be same as that of normal to the ...

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The coordinates of the point on the line through $(3,-4,-5)$ and $(2,-3,1)$ at which the line crosses the plane $2x+y+z=5$ is ?

answered Jan 2, 2014

Equation of the line through $P(3,-4,-5) \:\:and\:\:Q(2,-3,1)$ is given by$\large\frac{x-2}{1}=\...

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The distance of the point $(-1,-5,-10)$ from the point of intersection of the line $\overrightarrow r=(2\hat i-\hat j+2\hat k)+\lambda(3\hat i+4\hat j+12\hat k)$ and the plane $\overrightarrow r.(\hat i-\hat j+\hat k)=5$ is ?

answered Jan 2, 2014

Given eqn. of the line is written as $\large\frac{x-2}{3}=\frac{y+1}{4}=\frac{z-2}{12}=\lambda$Any...

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If from the point $P(a,b,c)$ perpendiculars $PA$ and $PB$ are drawn to $yz$ and $zx$ planes respectively then the vector equation of the plane $OAB$ is ?

answered Jan 2, 2014

Given that $A$ is foot of $\perp$ from $P(a,b,c)$ on $yz$ plane.$\therefore \;A=(0,b,c)$Simila...

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The equation of the plane whose foot of $\perp$ drawn from origin is $(1,2,1)$. is?

answered Jan 2, 2014

Since $A(1,2,1)$ is foot of $\perp$ from origin to the required plane, $A$ lies on the plane.Also ...

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The equation of the plane that meets the coordinated axis at the points $A,B,C$ respectively so that the centroid of the $\Delta ABC$ is $(\alpha,\beta,\gamma)$ is ?

answered Jan 1, 2014

Let the plane cut the coordinate axis at the points $A(a,0,0),\:B(0,b,0)\:and\:\:C(0,0,c)$$\there...

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The direction ratio of the line $3x+1=6y-2,\:z=1$ is ?

answered Jan 1, 2014

Convert the equation into standard form of the eqn, of line.$i.e.,$ $\large\frac{x+1/3}{1/3}=\frac...

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The point on the line $\overrightarrow r=(-2\hat i-\hat j+3\hat k)+\lambda(3\hat i+2\hat j+2\hat k)$ which is at a distance of $3\sqrt 2$ units from the point $(1,2,3)$ is ?

answered Jan 1, 2014

Given line is $\large\frac{x+2}{3}=\frac{y+1}{2}=\frac{z-3}{2}$$=\lambda$ Any point on the ...

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The equation of a line that are equally inclined with the coordinate axis and and which passes through $(-1,2,3)$ is?

answered Jan 1, 2014

Toolbox:$d.r.$ of the line making equal angle with coordinate axis is $(1,1,1)$$d.r.$ of the li...

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The angle between the lines whose $d.c.$ are given by the equations $3l+m+5n=0$ and $6mn-2nl+5lm=0$ is?

answered Jan 1, 2014

Toolbox:Angle between the two lines with d.r.$\overrightarrow b_1$ and $\overrightarrow b_2$ is ...

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The equation of the line through the point $(1,2,3)$ and $\perp$ to the plane $\overrightarrow r.(\hat i+2\hat j-5\hat k)+9=0$ is ?

answered Dec 31, 2013

Given eqn. of the plane is $\overrightarrow r.(\hat i+2\hat j-5\hat k)+9=0$Since the line is $\pe...

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The equation of the line through the point $(1,1,3)$ and parallel to the planes $\overrightarrow r.(\hat i-\hat j+2\hat k)=5$ and $\overrightarrow r.(3\hat i+\hat j+\hat k)=6$ is ?

answered Dec 31, 2013

Toolbox:$(\overrightarrow a\times\overrightarrow b)$ is a line (vector) $\perp$ to both $\overri...

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The line $\large\frac{x-2}{3}=\frac{y-3}{4}=\frac{z-4}{5}$ is parallel to the which of the following?

answered Dec 31, 2013

Toolbox:If the line is parallel to the plane then the line will be $\perp $ to its normal.Given line...

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If a line makes angle $\alpha,\:\beta,\:\gamma$ with $x-axis,\:y-axis\:and\:\:z-axis$ respectively, then $sin^2\alpha+sin^2\beta+sin^2\gamma=?$

answered Dec 31, 2013

Toolbox:$cos^2\alpha+cos^2\beta+cos^2\gamma=1$$sin^2\alpha+sin^2\beta+sin^2\gamma=1-cos^2\alpha+1-co...

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The angle between the planes $3x+4y+5z=3$ and $4x-3y+5z=2$ is ?

answered Dec 31, 2013

Toolbox:Angle between the two planes is $cos\theta=\large\frac{\overrightarrow n_1.\overrightarrow...

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The angle between the lines $\large\frac{2-x}{1}=\frac{y}{2}=\frac{z+3}{1}\:\:and\:\:\large\frac{x-4}{4}=\frac{y-1}{1}=\frac{z-5}{2}$ is?

answered Dec 31, 2013

Toolbox:Angle between the two lines is given by $cos\theta=\large\frac{(d.r.\:\:of\:line\:1).(d.r.\...

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If the planes $x=cy+bz,\:\:y=az+cx,\:z=bx+ay$ pass through a line then $a^2+b^2+c^2+2abc=?$

answered Dec 31, 2013

Given equations of the planes are $x-cy-bz=0......(i)$$cx-y+az=0..............(ii),\:\:\:\:and\:\:...

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If position vector of the points $A,B,C,D$ are given by $7\hat i-4\hat j+7\hat k,\:\hat i-6\hat j+10\hat k,\:-\hat i-3\hat j+4\hat k\:\:and\:\:5\hat i-\hat j+5\hat k$ respectively, then $ABCD$ is ?

answered Dec 30, 2013

$\overrightarrow {AB}=-6\hat i-2\hat j+3\hat k$, $\overrightarrow {AC}=-8\hat i+\hat j-3\hat k$, ...

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The length of median through $A$ of the $\Delta\:ABC$ where $\overrightarrow {AB}=3\hat i+4\hat k\:\:and\:\:\overrightarrow {AC}=5\hat i-2\hat j+4\hat k$ is ?

answered Dec 30, 2013

Toolbox:Median through $A$ of a $\Delta \:ABC$ is $\large\frac{\overrightarrow {AB}+\overrightarr...

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The shortest distance between the lines $\large\frac{x-6}{1}=\frac{2-y}{2}=\frac{z-2}{2}$ and $\large\frac{x+4}{3}=\frac{y}{-2}=\frac{1-z}{2}$ is?

answered Dec 29, 2013

Toolbox:S.D. between the lines $\overrightarrow r=\overrightarrow {a_1}+\lambda\overrightarrow { b_...

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The vector $\overrightarrow a=\alpha\hat i+2\hat j+\beta\hat k$ lies in the plane of the vectors $\overrightarrow b=\hat i+\hat j$ and $\overrightarrow c=\hat j+\hat k$ and bisects the angle between $\overrightarrow b\:\:and\:\:\overrightarrow c$, then $(\alpha,\beta)=?$

answered Dec 29, 2013

Given: $\overrightarrow a=\alpha \hat i+2\hat j+\beta \hat k,\:\overrightarrow b=\hat i+\hat j\:\:\...

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If the equation of the plane containing the line $\large\frac{x-x_1}{l}=\frac{y-y_1}{m}=\frac{z-z_1}{n}$ is $a(x-x_1)+b(y-y_1)+c(z-z_1)=0$, then ?

answered Dec 29, 2013

Toolbox:Every line on a plane is $\perp$ to the normal to the plane.If two lines are $\perp$, then...

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The direction ratio of the normal to the plane through $(1,0,0),(0,1,0)$, which makes $\large\frac{\pi}{4} $ with the plane $x+y=3$ is ?

answered Dec 28, 2013

Toolbox:Angle between two planes is given by $cos\theta=\large\frac{\overrightarrow n_1.\overrighta...

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The equation of the plane passing through the point $(3,2,0) $ and the line $\large\frac{x-3}{1}=\frac{y-6}{5}=\frac{z-4}{4}$ is ?

answered Dec 28, 2013

Let the equation of the plane be $ax+by+cz+d=0$$\Rightarrow\:$ The normal to the plane $\overrightar...

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