3D Geometry

The shortest distance between the lines \({{x - 3} \over 2} = {{y - 2} \over 3} = {{z - 1} \over { - 1}}\) and \({{x + 3} \over 2} = {{y - 6} \over 1} = {{z - 5} \over 3}\), is :

Let the foot of the perpendicular from the point (1, 2, 4) on the line \({{x + 2} \over 4} = {{y - 1} \over 2} = {{z + 1} \over 3}\) be P. Then the distance of P from the plane \(3x + 4y + 12z + 23 = 0\) is :

Let the line \(\frac{x-3}{7}=\frac{y-2}{-1}=\frac{z-3}{-4}\) intersect the plane containing the lines \(\frac{x-4}{1}=\frac{y+1}{-2}=\frac{z}{1}\) and \(4 a x-y+5 z-7 a=0=2 x-5 y-z-3, a \in \mathbb{R}\) at the point $$P(\alpha, \beta, \gamm...

If the plane \(P\) passes through the intersection of two mutually perpendicular planes \(2 x+k y-5 z=1\) and \(3 k x-k y+z=5, k<3\) and intercepts a unit length on positive \(x\)-axis, then the intercept made by the plane \(P\) on the $$y$...

If the line of intersection of the planes \(a x+b y=3\) and \(a x+b y+c z=0\), a \(>0\) makes an angle \(30^{\circ}\) with the plane \(y-z+2=0\), then the direction cosines of the line are :

If the length of the perpendicular drawn from the point \(P(a, 4,2)\), a \(>0\) on the line \(\frac{x+1}{2}=\frac{y-3}{3}=\frac{z-1}{-1}\) is \(2 \sqrt{6}\) units and \(Q\left(\alpha_{1}, \alpha_{2}, \alpha_{3}\right)\) is the image of the ...

Let the plane 2x + 3y + z + 20 = 0 be rotated through a right angle about its line of intersection with the plane x \(-\) 3y + 5z = 8. If the mirror image of the point \(\left( {2, - {1 \over 2},2} \right)\) in the rotated plane is B(a, b, ...

If the two lines \({l_1}:{{x - 2} \over 3} = {{y + 1} \over {-2}},\,z = 2\) and \({l_2}:{{x - 1} \over 1} = {{2y + 3} \over \alpha } = {{z + 5} \over 2}\) are perpendicular, then an angle between the lines l2 and $${l_3}:{{1 - x} \over 3} =...

Let \(\overrightarrow a = \widehat i + \widehat j + 2\widehat k\), \(\overrightarrow b = 2\widehat i - 3\widehat j + \widehat k\) and \(\overrightarrow c = \widehat i - \widehat j + \widehat k\) be three given vectors. Let $$\overrightar...

If the lines \(\overrightarrow r = \left( {\widehat i - \widehat j + \widehat k} \right) + \lambda \left( {3\widehat j - \widehat k} \right)\) and $$\overrightarrow r = \left( {\alpha \widehat i - \widehat j} \right) + \mu \left( {2\wideh...

If the plane \(2x + y - 5z = 0\) is rotated about its line of intersection with the plane \(3x - y + 4z - 7 = 0\) by an angle of \({\pi \over 2}\), then the plane after the rotation passes through the point :

Let \(\mathrm{Q}\) and \(\mathrm{R}\) be two points on the line \(\frac{x+1}{2}=\frac{y+2}{3}=\frac{z-1}{2}\) at a distance \(\sqrt{26}\) from the point \(P(4,2,7)\). Then the square of the area of the triangle \(P Q R\) is ___________.

The length of the perpendicular from the point \((1,-2,5)\) on the line passing through \((1,2,4)\) and parallel to the line \(x+y-z=0=x-2 y+3 z-5\) is :

The plane passing through the line \(L: l x-y+3(1-l) z=1, x+2 y-z=2\) and perpendicular to the plane \(3 x+2 y+z=6\) is \(3 x-8 y+7 z=4\). If \(\theta\) is the acute angle between the line \(L\) and the \(y\)-axis, then $$415 \cos ^{2} \th...

The largest value of \(a\), for which the perpendicular distance of the plane containing the lines \(\vec{r}=(\hat{i}+\hat{j})+\lambda(\hat{i}+a \hat{j}-\hat{k})\) and \(\vec{r}=(\hat{i}+\hat{j})+\mu(-\hat{i}+\hat{j}-a \hat{k})\) from the ...

A vector \(\vec{a}\) is parallel to the line of intersection of the plane determined by the vectors \(\hat{i}, \hat{i}+\hat{j}\) and the plane determined by the vectors \(\hat{i}-\hat{j}, \hat{i}+\hat{k}\). The obtuse angle between $$\vec{a...

Let the lines
\({L_1}:\overrightarrow r = \lambda \left( {\widehat i + 2\widehat j + 3\widehat k} \right),\,\lambda \in R\)
$${L_2}:\overrightarrow r = \left( {\widehat i + 3\widehat j + \widehat k} \right) + \mu \left( {\widehat i + \wi...

Let Q be the mirror image of the point P(1, 0, 1) with respect to the plane S : x + y + z = 5. If a line L passing through (1, \(-\)1, \(-\)1), parallel to the line PQ meets the plane S at R, then QR2 is equal to :

Let l1 be the line in xy-plane with x and y intercepts \({1 \over 8}\) and \({1 \over {4\sqrt 2 }}\) respectively, and l2 be the line in zx-plane with x and z intercepts \(- {1 \over 8}\) and \(- {1 \over {6\sqrt 3 }}\) respectively. If d...

Let p be the plane passing through the intersection of the planes \(\overrightarrow r \,.\,\left( {\widehat i + 3\widehat j - \widehat k} \right) = 5\) and $$\overrightarrow r \,.\,\left( {2\widehat i - \widehat j + \widehat k} \right) = 3$...

The line of shortest distance between the lines \(\frac{x-2}{0}=\frac{y-1}{1}=\frac{z}{1}\) and \(\frac{x-3}{2}=\frac{y-5}{2}=\frac{z-1}{1}\) makes an angle of \(\cos ^{-1}\left(\sqrt{\frac{2}{27}}\right)\) with the plane $$\mathrm{P}: \mat...

Let \(\mathrm{P}\) be the plane containing the straight line \(\frac{x-3}{9}=\frac{y+4}{-1}=\frac{z-7}{-5}\) and perpendicular to the plane containing the straight lines \(\frac{x}{2}=\frac{y}{3}=\frac{z}{5}\) and $$\frac{x}{3}=\frac{y}{7}=...

The shortest distance between the lines \(\frac{x+7}{-6}=\frac{y-6}{7}=z\) and \(\frac{7-x}{2}=y-2=z-6\) is :

A plane \(E\) is perpendicular to the two planes \(2 x-2 y+z=0\) and \(x-y+2 z=4\), and passes through the point \(P(1,-1,1)\). If the distance of the plane \(E\) from the point \(Q(a, a, 2)\) is \(3 \sqrt{2}\), then \((P Q)^{2}\) is equal...

If the shortest distance between the lines \(\overrightarrow r = \left( { - \widehat i + 3\widehat k} \right) + \lambda \left( {\widehat i - a\widehat j} \right)\) and $$\overrightarrow r = \left( { - \widehat j + 2\widehat k} \right) + \...

Let a line having direction ratios, 1, \(-\)4, 2 intersect the lines \({{x - 7} \over 3} = {{y - 1} \over { - 1}} = {{z + 2} \over 1}\) and \({x \over 2} = {{y - 7} \over 3} = {z \over 1}\) at the points A and B. Then (AB)2 is equal to ____...

Let the points on the plane P be equidistant from the points (\(-\)4, 2, 1) and (2, \(-\)2, 3). Then the acute angle between the plane P and the plane 2x + y + 3z = 1 is :

If the shortest distance between the lines \({{x - 1} \over 2} = {{y - 2} \over 3} = {{z - 3} \over \lambda }\) and \({{x - 2} \over 1} = {{y - 4} \over 4} = {{z - 5} \over 5}\) is \({1 \over {\sqrt 3 }}\), then the sum of all possible valu...

The square of the distance of the point of intersection of the line \({{x - 1} \over 2} = {{y - 2} \over 3} = {{z + 1} \over 6}\) and the plane \(2x - y + z = 6\) from the point (\(-\)1, \(-\)1, 2) is __________.

Let the equation of the plane, that passes through the point (1, 4, \(-\)3) and contains the line of intersection of the planes 3x \(-\) 2y + 4z \(-\) 7 = 0 and x + 5y \(-\) 2z + 9 = 0, be \(\alpha\)x + \(\beta\)y + \(\gamma\)z + 3 = 0, the...

Suppose, the line \({{x - 2} \over \alpha } = {{y - 2} \over { - 5}} = {{z + 2} \over 2}\) lies on the plane \(x + 3y - 2z + \beta = 0\). Then \((\alpha + \beta )\) is equal to _______.

The distance of the point (\(-\)1, 2, \(-\)2) from the line of intersection of the planes 2x + 3y + 2z = 0 and x \(-\) 2y + z = 0 is :

Let a plane P pass through the point (3, 7, \(-\)7) and contain the line, \({{x - 2} \over { - 3}} = {{y - 3} \over 2} = {{z + 2} \over 1}\). If distance of the plane P from the origin is d, then d2 is equal to ______________.

Let the plane passing through the point (\(-\)1, 0, \(-\)2) and perpendicular to each of the planes 2x + y \(-\) z = 2 and x \(-\) y \(-\) z = 3 be ax + by + cz + 8 = 0. Then the value of a + b + c is equal to :

The distance of the point P(3, 4, 4) from the point of intersection of the line joining the points. Q(3, \(-\)4, \(-\)5) and R(2, \(-\)3, 1) and the plane 2x + y + z = 7, is equal to ______________.

For real numbers \(\alpha\) and \(\beta\) \(\ne\) 0, if the point of intersection of the straight lines\({{x - \alpha } \over 1} = {{y - 1} \over 2} = {{z - 1} \over 3}\) and $${{x - 4} \over \beta } = {{y - 6} \over 3} = {{z - 7} \over 3}$...

Equation of a plane at a distance \(\sqrt {{2 \over {21}}}\) from the origin, which contains the line of intersection of the planes x \(-\) y \(-\) z \(-\) 1 = 0 and 2x + y \(-\) 3z + 4 = 0, is :

The distance of the point (1, \(-\)2, 3) from the plane x \(-\) y + z = 5 measured parallel to a line, whose direction ratios are 2, 3, \(-\)6 is :

Let S be the mirror image of the point Q(1, 3, 4) with respect to the plane 2x \(-\) y + z + 3 = 0 and let R(3, 5, \(\gamma\)) be a point of this plane. Then the square of the length of the line segment SR is ___________.

The equation of the plane passing through the line of intersection of the planes \(\overrightarrow r .\left( {\widehat i + \widehat j + \widehat k} \right) = 1\) and $$\overrightarrow r .\left( {2\widehat i + 3\widehat j - \widehat k} \righ...

The angle between the straight lines, whose direction cosines are given by the equations 2l + 2m \(-\) n = 0 and mn + nl + lm = 0, is :

Let (\(\lambda\), 2, 1) be a point on the plane which passes through the point (4, \(-\)2, 2). If the plane is perpendicular to the line joining the points (\(-\)2, \(-\)21, 29) and (\(-\)1, \(-\)16, 23), then $${\left( {{\lambda \over {11...

If (1, 5, 35), (7, 5, 5), (1, \(\lambda\), 7) and (2\(\lambda\), 1, 2) are coplanar, then the sum of all possible values of \(\lambda\) is :

Consider the three planesP1 : 3x + 15y + 21z = 9,P2 : x \(-\) 3y \(-\) z = 5, and P3 : 2x + 10y + 14z = 5Then, which one of the following is true?

If the mirror image of the point (1, 3, 5) with respect to the plane 4x \(-\) 5y + 2z = 8 is (\(\alpha\), \(\beta\), \(\gamma\)), then 5(\(\alpha\) + \(\beta\) + \(\gamma\)) equals :

Let L be a line obtained from the intersection of two planes x + 2y + z = 6 and y + 2z = 4. If point P(\(\alpha\), \(\beta\), \(\gamma\)) is the foot of perpendicular from (3, 2, 1) on L, then the value of 21(\(\alpha\) + \(\beta\) + $$\gam...

Let the line L be the projection of the line \({{x - 1} \over 2} = {{y - 3} \over 1} = {{z - 4} \over 2}\) in the plane x \(-\) 2y \(-\) z = 3. If d is the distance of the point (0, 0, 6) from L, then d2 is equal to _______________.

A plane P contains the line \(x + 2y + 3z + 1 = 0 = x - y - z - 6\), and is perpendicular to the plane \(- 2x + y + z + 8 = 0\). Then which of the following points lies on P?

Let Q be the foot of the perpendicular from the point P(7, \(-\)2, 13) on the plane containing the lines \({{x + 1} \over 6} = {{y - 1} \over 7} = {{z - 3} \over 8}\) and \({{x - 1} \over 3} = {{y - 2} \over 5} = {{z - 3} \over 7}\). Then (...

Let P be the plane passing through the point (1, 2, 3) and the line of intersection of the planes \(\overrightarrow r \,.\,\left( {\widehat i + \widehat j + 4\widehat k} \right) = 16\) and $$\overrightarrow r \,.\,\left( { - \widehat i + \w...