3D Geometry

Let L be the straight line joining the points P(1, 2, –1) and Q(2, 3, 1). Let S be the foot of the perpendicular drawn from the point R(4, –1, 5) to the line L. Another line passing through R intersects L at a point T such that the point S ...

Let P be the plane such that it contains the straight line $\frac{x-1}{2}=\frac{y-3}{3}=\frac{z+2}{1}$ and is perpendicular to the plane $x+2y+3z=4$. Let $P_1$ be the plane which passes through the point $(4,2,2)$ and is parallel to P.Then ...

Let $L_1$ be the line of intersection of the planes given by the equations$2x + 3y + z = 4$ and $x + 2y + z = 5$.Let $L_2$ be the line passing through the point $P(2, -1, 3)$ and parallel to $L_1$. Let $M$ denote the plane given by the equa...

A straight line drawn from the point $P(1,3,2)$, parallel to the line $\frac{x-2}{1}=\frac{y-4}{2}=\frac{z-6}{1}$, intersects the plane $L_1: x-y+3 z=6$ at the point $Q$. Another straight line which passes through $Q$ and is perpendicular t...

Let $\gamma \in \mathbb{R}$ be such that the lines $L_1: \frac{x+11}{1}=\frac{y+21}{2}=\frac{z+29}{3}$ and $L_2: \frac{x+16}{3}=\frac{y+11}{2}=\frac{z+4}{\gamma}$ intersect. Let $R_1$ be the point of intersection of $L_1$ and $L_2$. Let $O=...

Let $\mathbb{R}^3$ denote the three-dimensional space. Take two points $P=(1,2,3)$ and $Q=(4,2,7)$. Let $\operatorname{dist}(X, Y)$ denote the distance between two points $X$ and $Y$ in $\mathbb{R}^3$. Let

$$ \begin{gathered} S=\left\{X \i...

Let $\ell_1$ and $\ell_2$ be the lines $\vec{r}_1=\lambda(\hat{i}+\hat{j}+\hat{k})$ and $\vec{r}_2=(\hat{j}-\hat{k})+\mu(\hat{i}+\hat{k})$, respectively. Let $X$ be the set of all the planes $H$ that contain the line $\ell_1$. For a plane $...

Let \(S\) be the reflection of a point \(Q\) with respect to the plane given by

\(\vec{r}=-(t+p) \hat{\imath}+t \hat{\jmath}+(1+p) \hat{k}\)

where \(t, p\) are real parameters and \(\hat{\imath}, \hat{\jmath}, \hat{k}\) are the unit vec...

Let \(P_{1}\) and \(P_{2}\) be two planes given by

$$ \begin{aligned} &P_{1}: 10 x+15 y+12 z-60=0 \\\\ &P_{2}:-2 x+5 y+4 z-20=0 \end{aligned} $$

Which of the following straight lines can be an edge of some tetrahedron whose two faces lie ...

Let \(\alpha\)2 + \(\beta\)2 + \(\gamma\)2 \(\ne\) 0 and \(\alpha\) + \(\gamma\) = 1. Suppose the point (3, 2, \(-\)1) is the mirror image of the point (1, 0, \(-\)1) with respect to the plane \(\alpha\)x + \(\beta\)y + \(\gamma\)...

Let L1 and L2 be the following straight lines.\({L_1}:{{x - 1} \over 1} = {y \over { - 1}} = {{z - 1} \over 3}\) and \({L_2}:{{x - 1} \over { - 3}} = {y \over { - 1}} = {{z - 1} \over 1}\).Suppose the straight line $$L:{{x - \alpha } \over ...

Three lines \({L_1}:r = \lambda \widehat i\), \(\lambda\) \(\in\) R,\({L_2}:r = \widehat k + \mu \widehat j\), \(\mu\) \(\in\) R and \({L_3}:r = \widehat i + \widehat j + v\widehat k\), v \(\in\) R are given.For which point(s) Q on ...

Let L1 and L2 denote the lines\(r = \widehat i + \lambda ( - \widehat i + 2\widehat j + 2\widehat k)\), \(\lambda\)\(\in\) R
and \(r = \mu (2\widehat i - \widehat j + 2\widehat k),\,\mu \in R\) respectively. If L3 is a line which is pe...

Three lines are given by \(r = \lambda \widehat i,\,\lambda \in R\), \(r = \mu (\widehat i + \widehat j),\,\mu \in R\) and \(r = v(\widehat i + \widehat j + \widehat k),\,v\, \in R\)Let the lines cut the plane x + y + z = 1 at the points ...

Let P be a point in the first octant, whose image Q in the plane x + y = 3 (that is, the line segment PQ is perpendicular to the plane x + y = 3 and the mid-point of PQ lies in the plane x + y = 3) lies on the Z-axis. Let the distance of P ...

Consider the cube in the first octant with sides OP, OQ and OR of length 1, along the X-axis, Y-axis and Z-axis, respectively, where O(0, 0, 0) is the origin. Let \(S\left( {{1 \over 2},{1 \over 2},{1 \over 2}} \right)\) be the centre of th...

Let P1 : 2x + y \(-\) z = 3 and P2 : x + 2y + z = 2 be two planes. Then, which of the following statement(s) is(are) TRUE?

The equation of the plane passing through the point (1, 1, 1) and perpendicular to the planes 2x + y \(-\) 2z = 5 and 3x \(-\) 6y \(-\) 2z = 7 is

Let \(P\) be the image of the point \((3,1,7)\) with respect to the plane \(x-y+z=3.\) Then the equation of the plane passing through \(P\) and containing the straight line \({x \over 1} = {y \over 2} = {z \over 1}\) is

Consider a pyramid \(OPQRS\) located in the first octant \(\left( {x \ge 0,y \ge 0,z \ge 0} \right)\) with \(O\) as origin, and \(OP\) and \(OR\) along the \(x\)-axis and the \(y\)-axis, respectively. The base \(OPQR\) of the pyramid is a s...

In \({R^3},\) let \(L\) be a straight lines passing through the origin. Suppose that all the points on \(L\) are at a constant distance from the two planes \({P_1}:x + 2y - z + 1 = 0\) and \({P_2}:2x - y + z - 1 = 0.\) Let \(M\) be the loc...

In \({R^3},\) consider the planes \(\,{P_1}:y = 0\) and \({P_2}:x + z = 1.\) Let \({P_3}\) be the plane, different from \({P_1}\) and \({P_2}\), which passes through the intersection of \({P_1}\) and \({P_2}.\) If the distance of the point ...

From a point \(P\left( {\lambda ,\lambda ,\lambda } \right),\) perpendicular \(PQ\) and \(PR\) are drawn respectively on the lines \(y=x, z=1\) and \(y=-x, z=-1.\) If \(P\) is such that \(\angle QPR\) is a right angle, then the possible val...

Consider the lines \({L_1}:{{x - 1} \over 2} = {y \over { - 1}} = {{z + 3} \over 1},{L_2} : {{x - 4} \over 1} = {{y + 3} \over 1} = {{z + 3} \over 2}\) and the planes \({P_1}:7x + y + 2z = 3,{P_2} = 3x + 5y - 6z = 4.\) Let \(ax+by+cz=d\) be...

Two lines \({L_1}:x = 5,{y \over {3 - \alpha }} = {z \over { - 2}}\) and \({L_2}:x = \alpha ,{y \over { - 1}} = {z \over {2 - \alpha }}\) are coplanar. Then \(\alpha\) can take value(s)

A line \(l\) passing through the origin is perpendicular to the lines
\($\,{l_1}:\left( {3 + t} \right)\widehat i + \left( { - 1 + 2t} \right)\widehat j + \left( {4 + 2t} \right)\widehat k,\,\,\,\,\, - \infty < t < \infty\)$
$$${l_2}:\le...

Perpendiculars are drawn from points on the line $\frac{x+2}{2}=\frac{y+1}{-1}=\frac{z}{3}$ to the plane $x+y+$ $z=3$. The foot of perpendiculars lie on the line

If the straight lines \(\,{{x - 1} \over 2} = {{y + 1} \over k} = {z \over 2}\) and \({{x + 1} \over 5} = {{y + 1} \over 2} = {z \over k}\) are coplanar, then the plane (s) containing these two lines is (are)

The equation of a plane passing through the line of intersection of the planes \(x+2y+3z=2\) and \(x-y+z=3\) and at a distance \({2 \over {\sqrt 3 }}\) from the point \((3, 1, -1)\) is

The point \(P\) is the intersection of the straight line joining the points \(Q(2, 3, 5)\) and \(R(1, -1, 4)\) with the plane \(5x-4y-z=1.\) If \(S\) is the foot of the perpendicular drawn from the point \(T(2, 1, 4)\) to \(QR,\) then the l...

Match the statement in Column-\(I\) with the values in Column-\(II\)
\(\,\,\,\,\) \(\,\,\,\,\) \(\,\,\,\,\) Column-\(I\)
(A)\(\,\,\,\,\) A line from the origin meets the lines $$\,{{x - 2} \over 1} = {{y - 1} \over { - 2}} = {{z + 1} \ov...

If the distance of the point \(P(1, -2, 1)\) from the plane \(x+2y-2z\)\(\, = \alpha ,\) where \(\alpha > 0,\) is \(5,\) then the foot of the perpendicular from \(P\) to the planes is

Equation of the plane containing the straight line \({x \over 2} = {y \over 3} = {z \over 4}\) and perpendicular to the plane containing the straight lines \({x \over 3} = {y \over 4} = {z \over 2}\) and $${x \over 4} = {y \over 2} = {z \ov...

If the distance between the plane \(Ax-2y+z=d\) and the plane containing the lines \({{x - 1} \over 2} = {{y - 2} \over 3} = {{z - 3} \over 4}\) and \({{x - 2} \over 3} = {{y - 3} \over 4} = {{z - 4} \over 5}\,\) is \(\sqrt 6 \,\,,\) then...

A line with positive direction cosines passes through the point P(2, \(-\)1, 2) and makes equal angles with the coordinate axes. The line meets the plane \(2x + y + z = 9\) at point Q. The length of the line segment PQ equals

Let \(P(3,2,6)\) be a point in space and \(Q\) be a point on the line
\($\widehat r = \left( {\widehat i - \widehat j + 2\widehat k} \right) + \mu \left( { - 3\widehat i + \widehat j + 5\widehat k} \right)\)$
Then the value of \(\mu\) for...

The distance of the point \((1, 1, 1)\) from the plane passing through the point \((-1, -2, -1)\) and whose normal is perpendicular to both the lines \({L_1}\) and \({L_2}\) is :

Consider three planes
\(${P_1}:x - y + z = 1\)$
\(${P_2}:x + y - z = 1\)$
\(${P_3}:x - 3y + 3z = 2\)$
Let \({L_1},\) \({L_2},\) \({L_3}\) be the lines of intersection of the planes \({P_2}\) and \({P_3},\) \({P_3}\) and \({P_1},\) $${P...

Consider the planes \(3 x-6 y-2 z=15\) and \(2 x+y-2 z=5\).
STATEMENT - 1 : The parametric equations of the line of intersection of the given planes are \(x=3+14 t, y=1+2 t, z=15 t\)
STATEMENT - 2 : The vectors $$14 \hat{i}+2 \hat{j}+15 \ha...

Consider the following linear equations \(ax+by+cz=0;\) \(\,\,\,\) \(bx+cy+az=0;\) \(\,\,\,\) \(cx+ay+bz=0\)
Match the conditions/expressions in Column \(I\) with statements in Column \(II\) and indicate your answer by darkening the approp...

Consider the planes \(3x-6y-2z=15\) and \(2x+y-2z=5.\)
STATEMENT-1: The parametric equations of the line of intersection of the given planes are \(x=3+14t,y=1+2t,z=15t.\) because
STATEMENT-2: The vector $${14\widehat i + 2\widehat j + 15\w...

A plane passes through $(1,-2,1)$ and is perpendicular to two planes $2 x-2 y+z=0$ and $x-y+2 z=4$. The distance of the plane from the point $(1,2,2)$ is:

Let \({\overrightarrow A }\) be vector parallel to line of intersection of planes \({P_1}\) and \({P_2}.\) Planes \({P_1}\) is parallel to the vectors \(2\widehat j + 3\widehat k\) and \(4\widehat j - 3\widehat k\) and that \({P_2}\) is pa...

Match the following:

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.tg td{border-color:black;border-style:solid;border-width:1px;font-family:Arial, sans-serif;font-size:14px;
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A variable plane at a distance of the one unit from the origin cuts the coordinates axes at \(A,\) \(B\) and \(C.\) If the centroid \(D\) \((x, y, z)\) of triangle \(ABC\) satisfies the relation $${1 \over {{x^2}}} + {1 \over {{y^2}}} + {1 ...

Find the equation of the plane containing the line \(2 x-y+z-3=0,3 x+y+z=5\) and at a distance of \(\frac{1}{\sqrt{6}}\) from the point \((2,1,-1)\).

Find the equation of the plane containing the line \(2x-y+z-3=0,3x+y+z=5\) and at a distance of \({1 \over {\sqrt 6 }}\) from the point \((2, 1, -1).\)

If the lines \({{x - 1} \over 2} = {{y + 1} \over 3} = {{z - 1} \over 4}\) and \(\,{{x - 3} \over 1} = {{y - k} \over 2} = {z \over 1}\) intersect, then the value of \(k\) is

Find the equation of plane passing through \((1, 1, 1)\) & parallel to the lines \({L_1},{L_2}\) having direction ratios \((1,0,-1),(1,-1,0).\) Find the volume of tetrahedron formed by origin and the points where these planes intersect the ...

A parallelopiped \('S'\) has base points \(A, B, C\) and \(D\) and upper face points \(A',\) \(B',\) \(C'\) and \(D'.\) This parallelopiped is compressed by upper face \(A'B'C'D'\) to form a new parallelopiped \('T'\) having upper face poin...