Three Dimensional Geometry
Free NCERT & Exemplar step-by-step solutions — CBSE Class 12 Mathematics
NCERT Exemplar
Q37
If a plane passes through the points (2,0,0) (0,3,0) and (0,0,4) the equation of plane is …………
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Q38
The direction cosines of the vector $(2\widehat {\rm{i}} + 2\widehat {\rm{j}} - \widehat {\rm{k}})$ are ………….
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Q39
The vector equation of the line $\frac{{x - 5}}{3} = \frac{{y + 4}}{7} = \frac{{z - 6}}{2}$ is ………...
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Q40
The vector equation of the line through the points (3,4,-7) and (1, -1, 6) is………..
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Q41
The cartesian equation of the plane $\overrightarrow {\rm{r}} \cdot (\widehat {\rm{i}} + \widehat {\rm{j}} - \widehat {\rm{k}}) = 2$ is ………...
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Q16
. Find the foot of perpendicular from the point (2,3,-8) to the line $\frac{{4 - x}}{2} = \frac{y}{6} = \frac{{1 - z}}{3}$. Also, find the perpendicular distance from the given point to the line.
LA
Q17
Find the distance of a point (2,4,-1) from the line $\frac{{x + 5}}{1} = \frac{{y + 3}}{4} = \frac{{z - 6}}{{ - 9}}$
LA
Q18
Find the length and the foot of perpendicular from the point $\left( {1,\frac{3}{2},2} \right)$ to the plane $2x - 2y + 4z + 5 = 0$.
LA
Q19
Find the equation of the line passing through the point (3,0,1) and parallel to the planes $x + 2y = 0$ and $3y - z = 0$.
LA
Q20
Find the equation of the plane through the points (2,1,-1),(-1,3,4) and perpendicular to the plane $x - 2y + 4z = 10$.
LA
Q21
Find the shortest distance between the lines gives by
$\overrightarrow {\rm{r}} = (8 + 3\lambda )\widehat {\rm{i}} - (9 + 16\lambda )\widehat {\rm{j}} + (10 + 7\lambda )\widehat {\rm{k}}$
$\overrightarrow {\rm{r}} = 3\widehat {\rm{i}} - 2\widehat {\rm{j}} + 6\widehat {\rm{k}} + \lambda (2\widehat {\rm{i}} + \widehat {\rm{j}} + 2\widehat {\rm{k}})$
$= {p^\prime }y + {q^\prime },z = {r^\prime }y + {s^\prime }$ are perpendicular, if $p{p^\prime } + r{r^\prime } + 1 = 0$. SA Q7 Find the equation of a plane which bisects perpendicularly the line joining the points $A(2,3,4)$ and $B(4,5,8)$ at right angles. SA Q8 Find the equation of a plane which is at a distance $3\sqrt 3$ units from origin and the normal to which is equally inclined to coordinate axis. SA Q9 If the line drawn from the point $\left( { - 2, - 1, - 3} \right)$ meets a plane at right angle at the point $(1, - 3,3)$, then find the equation of the plane. SA Q10 Find the equation of the plane through the points (2,1,0),(3,-2,-2) and (3,1,7) SA Q11 Find the equations of the two lines through the origin which intersect the line $\frac{{x - 3}}{2} = \frac{{y - 3}}{1} = \frac{z}{1}$ at angles of $\frac{\pi }{3}$ each. SA Q12 Find the angle between the lines whose direction cosines are given by the equation $l + m + n = 0$ and ${l^2} + {m^2} - {n^2} = 0$. SA Q13 If a variable line in two adjacent positions has direction cosines $l,m,n$ and $l + \delta l,m + \delta m,n + \delta n$, then show that the small angle $\delta \theta$ between the two positions is given by $\delta {\theta ^2} = \delta {l^2} + \delta {m^2} + \delta {n^2}$. SA Q14 If 0 is the origin and $A$ is $(a,b,c)$, then find the direction cosines of the line $OA$ and the equation of plane through $A$ at right angle to $OA$. SA Q15 Two systems of rectangular axis have the same origin. If a plane cuts them at distances $a,b,c$ and ${a^\prime },{b^\prime },{c^\prime }$, respectively from the origin, then prove that $\frac{1}{{{a^2}}} + \frac{1}{{{b^2}}} + \frac{1}{{{c^2}}} = \frac{1}{{{a^{\prime 2}}}} + \frac{1}{{{b^{\prime 2}}}} + \frac{1}{{{c^{\prime 2}}}}$. SA Q25 Prove that the line through $A(0, - 1, - 1)$ and $B(4,5,1)$ intersects the line through $C(3,9,4)$ and $D( - 4,4,4)$. SA
$\overrightarrow {\rm{r}} = (8 + 3\lambda )\widehat {\rm{i}} - (9 + 16\lambda )\widehat {\rm{j}} + (10 + 7\lambda )\widehat {\rm{k}}$
and $\overrightarrow {\rm{r}} = 15\widehat {\rm{i}} + 29\widehat {\rm{j}} + 5\widehat {\rm{k}} + \mu (3\widehat {\rm{i}} + 8\widehat {\rm{j}} - 5\widehat {\rm{k}})$
LA Q22 Find the equation of the plane which is perpendicular to the plane $5x + 3y + 6z + 8 = 0$ and which contains the line of intersection of the planes $x + 2y + 3z - 4 = 0$ and $2x + y - z + 5 = 0$. LA Q23 If the plane $ax + by = 0$ is rotated about its line of intersection with the plane $z = 0$ through an angle $\alpha$, then prove that the equation of the plane in its new position is $ax + by \pm \left( {\sqrt {{a^2} + {b^2}} \tan \alpha } \right)z = 0$. LA Q24 Find the equation of the plane through the intersection of the planes $\overrightarrow {\rm{r}} \cdot (\widehat {\rm{i}} + 3\widehat {\rm{j}}) - 6 = 0$ and $\overrightarrow {\rm{r}} \cdot (3\widehat {\rm{i}} - \widehat {\rm{j}} - 4\widehat {\rm{k}}) = 0,$ whose perpendicular distance from origin is unity. LA Q25 Show that the points $(\widehat {\rm{i}} - \widehat {\rm{j}} + 3\widehat {\rm{k}})$ and $3(\widehat {\rm{i}} + \widehat {\rm{j}} + \widehat {\rm{k}})$ are equidistant from the plane $\overrightarrow {\rm{r}} \cdot (5\widehat {\rm{i}} + 2\widehat {\rm{j}} - 7\widehat {\rm{k}}) + 9 = 0$ and lies on opposite side of it. LA Q26 $\overrightarrow {{\rm{AB}}} = 3\widehat {\rm{i}} - \widehat {\rm{j}} + \widehat {\rm{k}}$and $\overrightarrow {{\rm{CD}}} = - 3\widehat {\rm{i}} + 2\widehat {\rm{j}} + 4\widehat {\rm{k}}$ are two vectors.z The position vectors of the points $A$ and $C$ are $6\widehat {\rm{i}} + 7\widehat {\rm{j}} + 4\widehat {\rm{k}}$ and $- 9\widehat {\rm{i}} + 2\widehat {\rm{k}}$, respectively. Find the position vector of a point $P$ on the line $A B$ and a point $Q$ on the line $CD$ such that $\overrightarrow {{\rm{PQ}}}$ is perpendicular to $\overrightarrow {{\rm{AB}}}$ and $\overrightarrow {{\rm{CD}}}$ both.
LA Q27 Show that the straight lines whose direction cosines are given by $2l + 2m - n = 0$ and $mn + nl + lm = 0$ are at right angles. LA Q28 If ${l_1},{m_1},{n_1},{l_2},{m_2},{n_2}$ and ${l_3},{m_3},{n_3}$ are the direction cosines of three mutually perpendicular lines, then prove that the line whose direction cosines are proportional to ${l_1} + {l_2} + {l_3},{m_1} + {m_2} + {m_3}$ and ${n_1} + {n_2} + {n_3}$ makes equal angles with them. LA Q29 Distance of the point $(\alpha ,\beta ,\gamma )$ from Y-axis is MCQ Q30 If the direction cosines of a line are $k$, $k$ and $k$, then MCQ Q31 The distance of the plane $\overrightarrow {\rm{r}} \left( {\frac{2}{7}\widehat {\rm{i}} + \frac{3}{7}\widehat {\rm{j}} - \frac{6}{7}\widehat {\rm{k}}} \right) = 1$ from the origin is MCQ Q32 The sine of the angle between the straight line $\frac{{x - 2}}{3} = \frac{{y - 3}}{4} = \frac{{z - 4}}{5}$ and the plane $2x - 2y + z = 5$ is MCQ Q33 The reflection of the point $(\alpha ,\beta ,\gamma )$ in the XY-plane is MCQ Q34 The area of the quadrilateral ABCD where $A(0,4,1)$ $B(2,3, - 1),C(4,5,0)$, and $D(2,6,2)$ is equal to MCQ Q35 The locus represented by $xy + yz = 0$ is MCQ Q36 If the plane $2x - 3y + 6z - 11 = 0$ makes an angle ${\sin ^{ - 1}}\alpha$ with $X$-axis, then the value of $\alpha$ is MCQ Q1 Find the position vector of a point $A$ in space such that $\overrightarrow {{\rm{OA}}}$ is inclined at ${60^\circ }$ to OX and at ${45^\circ }$ to OY and $|\overrightarrow {{\rm{OA}}} | = 10$ units. SA Q2 Find the vector equation of the line which is parallel to the vector $3\widehat {\rm{i}} - 2\widehat {\rm{j}} + 6\widehat {\rm{k}}$ and which passes through the point (1,-2,3) SA Q3 Show that the lines $\frac{{x - 1}}{2} = \frac{{y - 2}}{3} = \frac{{z - 3}}{4}$ and $\frac{{x - 4}}{5} = \frac{{y - 1}}{2} = z$ intersect. Also, find their point of intersection.If shortest distance between the lines is zero, then they intersect.
SA Q4 Find the angle between the lines$\overrightarrow {\rm{r}} = 3\widehat {\rm{i}} - 2\widehat {\rm{j}} + 6\widehat {\rm{k}} + \lambda (2\widehat {\rm{i}} + \widehat {\rm{j}} + 2\widehat {\rm{k}})$
and $\overrightarrow {\rm{r}} = (2\widehat {\rm{j}} - 5\widehat {\rm{k}}) + \mu (6\widehat {\rm{i}} + 3\widehat {\rm{j}} + 2\widehat {\rm{k}})$
As we know, $\cos \theta = \frac{{\left| {{{\overrightarrow {\rm{b}} }_1} \cdot {{\overrightarrow {\rm{b}} }_2}} \right|}}{{|\overrightarrow {\rm{b}} | \cdot \left| {{{\overrightarrow {\rm{b}} }_2}} \right|}}$
where, $\theta$ is the angle between the lines ${\overrightarrow {\rm{a}} _1} + \lambda {\overrightarrow {\rm{b}} _1}$ and $\overrightarrow {{{\rm{a}}_2}} + \mu \overrightarrow {{{\rm{b}}_2}}$.
SA Q6 Prove that the lines $x = py + q,z = ry + s$ and $x$$= {p^\prime }y + {q^\prime },z = {r^\prime }y + {s^\prime }$ are perpendicular, if $p{p^\prime } + r{r^\prime } + 1 = 0$. SA Q7 Find the equation of a plane which bisects perpendicularly the line joining the points $A(2,3,4)$ and $B(4,5,8)$ at right angles. SA Q8 Find the equation of a plane which is at a distance $3\sqrt 3$ units from origin and the normal to which is equally inclined to coordinate axis. SA Q9 If the line drawn from the point $\left( { - 2, - 1, - 3} \right)$ meets a plane at right angle at the point $(1, - 3,3)$, then find the equation of the plane. SA Q10 Find the equation of the plane through the points (2,1,0),(3,-2,-2) and (3,1,7) SA Q11 Find the equations of the two lines through the origin which intersect the line $\frac{{x - 3}}{2} = \frac{{y - 3}}{1} = \frac{z}{1}$ at angles of $\frac{\pi }{3}$ each. SA Q12 Find the angle between the lines whose direction cosines are given by the equation $l + m + n = 0$ and ${l^2} + {m^2} - {n^2} = 0$. SA Q13 If a variable line in two adjacent positions has direction cosines $l,m,n$ and $l + \delta l,m + \delta m,n + \delta n$, then show that the small angle $\delta \theta$ between the two positions is given by $\delta {\theta ^2} = \delta {l^2} + \delta {m^2} + \delta {n^2}$. SA Q14 If 0 is the origin and $A$ is $(a,b,c)$, then find the direction cosines of the line $OA$ and the equation of plane through $A$ at right angle to $OA$. SA Q15 Two systems of rectangular axis have the same origin. If a plane cuts them at distances $a,b,c$ and ${a^\prime },{b^\prime },{c^\prime }$, respectively from the origin, then prove that $\frac{1}{{{a^2}}} + \frac{1}{{{b^2}}} + \frac{1}{{{c^2}}} = \frac{1}{{{a^{\prime 2}}}} + \frac{1}{{{b^{\prime 2}}}} + \frac{1}{{{c^{\prime 2}}}}$. SA Q25 Prove that the line through $A(0, - 1, - 1)$ and $B(4,5,1)$ intersects the line through $C(3,9,4)$ and $D( - 4,4,4)$. SA
Exercise 11.1
Q1
If a line makes angles 90°, 135°, 45° with the x, y and z-axes respectively, find its direction cosines.
SA
Q2
Find the direction cosines of a line which makes equal angles with the coordinate axes.
SA
Q3
If a line has direction ratios $- 18,12, - 4$ then what are its direction cosines ?
SA
Q4
Show that the points $(2,3,4),( - 1, - 2,1),(5,8,7)$ are collinear.
SA
Q5
Find the direction cosines of the sides of the triangle whose vertices are$(3,5, - 4),( - 1,1,2)$ and $( - 5, - 5, - 2)$ and (- 5, - 5, - 2)
SA
Exercise 11.2
Q1
Show that the three lines with direction cosines
Write its vector form. SA Q8 Find the vector and the cartesian equations of the line that passes through the origin and $(5, - 2,3)$. SA Q9 Find the vector and the cartesian equations of the line that passes through the points $(3, - 2, - 5),(3, - 2,6)$ . SA Q10 Find the angle between the following pairs of lines:
(i) $\vec r = 2\hat i - 5\hat j + \hat k + \lambda (3\hat i + 2\hat j + 6\hat k)$ and
$\vec r = 7\hat i - 6\hat k + \mu (\hat i + 2\hat j + 2\hat k)$
(i) $\cfrac{{x - 2}}{2} = \cfrac{{y - 1}}{5} = \cfrac{{z + 3}}{{ - 3}}$ and $\cfrac{{x + 2}}{{ - 1}} = \cfrac{{y - 4}}{8} = \cfrac{{z - 5}}{4}$
(ii) $\cfrac{x}{2} = \cfrac{y}{2} = \cfrac{z}{1}$ and $\cfrac{{x - 5}}{4} = \cfrac{{y - 2}}{1} = \cfrac{{z - 3}}{8}$ SA Q12 Find the values of p so that the lines $\cfrac{{1 - x}}{3} = \cfrac{{7y - 14}}{{2p}} = \cfrac{{z - 3}}{2}$
and $\cfrac{{7 - 7x}}{{3p}} = \cfrac{{y - 5}}{1} = \cfrac{{6 - z}}{5}$ are at right angles. SA Q13 Show that the lines $\cfrac{{x - 5}}{7} = \cfrac{{y + 2}}{{ - 5}} = \cfrac{z}{1}$ and $\cfrac{x}{1} = \cfrac{y}{2} = \cfrac{z}{3}$ are perpendicular to each other. SA Q14 Find the shortest distance between the lines
$\vec r = (\hat i + 2\hat j + \hat k) + \lambda (\hat i - \hat j + \hat k)$ and
$\vec r = 2\hat i - \hat j - \hat k + \mu (2\hat i + \hat j + 2\hat k)$ SA Q15 Find the shortest distance between the lines
$\cfrac{{12}}{{13}},\cfrac{{ - 3}}{{13}},\cfrac{{ - 4}}{{13}};\cfrac{4}{{13}},\cfrac{{12}}{{13}},\cfrac{3}{{13}};\cfrac{3}{{13}},\cfrac{{ - 4}}{{13}},\cfrac{{12}}{{13}}$ are mutually perpendicular.
SA Q2 Show that the line through the points $(1, - 1,2),(3,4, - 2)$ is perpendicular to the line through the points $(0,3,2)$ and $(3,5,6)$. SA Q3 Show that the line through the points $(4,7,8),(2,3,4)$ is parallel to the line through the points $( - 1, - 2,1),(1,2,5).$ SA Q4 Find the equation of the line which passes through the point $(1,2,3)$ and is parallel to the vector$3\hat i + 2\hat j - 2\hat k$ . SA Q5 Find the equation of the line in vector and in cartesian form that passes through the point with position vector $2\hat i - \hat j + 4\hat k$ and is in the direction $\hat i + 2\hat j - \hat k.$ SA Q6 Find the cartesian equation of the line which passes through the point $( - 2,4, - 5)$ and is parallel to the line given by $\cfrac{{x + 3}}{3} = \cfrac{{y - 4}}{5} = \cfrac{{z + 8}}{6}$. SA Q7 The cartesian equation of a line is $\cfrac{{x - 5}}{3} = \cfrac{{y + 4}}{5} = \cfrac{{z + 5}}{6}$ .Write its vector form. SA Q8 Find the vector and the cartesian equations of the line that passes through the origin and $(5, - 2,3)$. SA Q9 Find the vector and the cartesian equations of the line that passes through the points $(3, - 2, - 5),(3, - 2,6)$ . SA Q10 Find the angle between the following pairs of lines:
(i) $\vec r = 2\hat i - 5\hat j + \hat k + \lambda (3\hat i + 2\hat j + 6\hat k)$ and
$\vec r = 7\hat i - 6\hat k + \mu (\hat i + 2\hat j + 2\hat k)$
(ii) $\vec r = 3\hat i + \hat j - 2\hat k + \lambda (\hat i - \hat j - 2\hat k)$ and
$\vec r = 2\hat i - \hat j - 56\hat k + \mu (3\hat i - 5\hat j - 4\hat k)$
(i) $\cfrac{{x - 2}}{2} = \cfrac{{y - 1}}{5} = \cfrac{{z + 3}}{{ - 3}}$ and $\cfrac{{x + 2}}{{ - 1}} = \cfrac{{y - 4}}{8} = \cfrac{{z - 5}}{4}$
(ii) $\cfrac{x}{2} = \cfrac{y}{2} = \cfrac{z}{1}$ and $\cfrac{{x - 5}}{4} = \cfrac{{y - 2}}{1} = \cfrac{{z - 3}}{8}$ SA Q12 Find the values of p so that the lines $\cfrac{{1 - x}}{3} = \cfrac{{7y - 14}}{{2p}} = \cfrac{{z - 3}}{2}$
and $\cfrac{{7 - 7x}}{{3p}} = \cfrac{{y - 5}}{1} = \cfrac{{6 - z}}{5}$ are at right angles. SA Q13 Show that the lines $\cfrac{{x - 5}}{7} = \cfrac{{y + 2}}{{ - 5}} = \cfrac{z}{1}$ and $\cfrac{x}{1} = \cfrac{y}{2} = \cfrac{z}{3}$ are perpendicular to each other. SA Q14 Find the shortest distance between the lines
$\vec r = (\hat i + 2\hat j + \hat k) + \lambda (\hat i - \hat j + \hat k)$ and
$\vec r = 2\hat i - \hat j - \hat k + \mu (2\hat i + \hat j + 2\hat k)$ SA Q15 Find the shortest distance between the lines
$\cfrac{{x + 1}}{7} = \cfrac{{y + 1}}{{ - 6}} = \cfrac{{z + 1}}{1}$ and $\cfrac{{x - 3}}{1} = \cfrac{{y - 5}}{{ - 2}} = \cfrac{{z - 7}}{1}$
SA Q16 Find the shortest distance between the lines whose vector equations are $\vec r = (\hat i + 2\hat j + 3\hat k) + \lambda (\hat i - 3\hat j + 2\hat k)$ and $\vec r = 4\hat i + 5\hat j + 6\hat k + \mu (2\hat i + 3\hat j + \hat k)$ SA Q17 Find the shortest distance between the lines whose vector equations are $\vec r = (1 - t)\hat i + (t - 2)\hat j + (3 - 2t)\hat k$ and $\vec r = (s + 1)\hat i + (2s - 1)\hat j - (2s + 1)\hat k$ SAExercise 11.3
Q1
In each of the following cases, determine the direction cosines of the normal to the plane and the distance from the origin.
(a) $z = 2$
(b) $x + y + z = 1$
(c) $2x + 3y - z = 5$
(d) $5y + 8 = 0$
(a) $\vec r \cdot (\hat i + \hat j - \hat k) = 2$
(b) $\vec r \cdot (2\hat i + 3\hat j - 4\hat k) = 1$
(c)$\vec r \cdot [(s - 2t)\hat i + (3 - t)\hat j + (2s + t)\hat k] = 15$,
(a) $2x + 3y + 4z - 12 = 0$
(b) $3y + 4z - 6 = 0$
(c)$x + y + z = 1$
(d) $5y + 8 = 0$
(a) that passes through the point $(1,0, - 2)$ and the normal to the plane is $(\hat i + \hat j - \hat k)$ .
(b) that passes through the point $(1,4,6)$and the normal vector to the plane is $\hat i - 2\hat j + \hat k$
(a) $\left( {1,1, - 1} \right),\left( {6,4, - 5} \right),\left( { - 4, - 2,3} \right)$
(b) $\left( {1,1,0} \right),\left( {1,2,1} \right),\left( { - 2,2, - 1} \right)$
SA Q7 Find the intercepts cut off by the plane$2x + y - z = 5.$ SA Q8 Find the equation of the plane with intercept 3 on the y-axis and parallel to ZOX plane. SA Q9 Find the equation of the plane through the intersection of the planes $3x - y + 2z - 4 = 0$ and $x + y + z - 2 = 0$ and the point$(2,2,1)$. SA Q10 Find the vector equation of the plane passing through the intersection of the planes $\vec r \cdot (2\hat i + 2\hat j - 3\hat k) = 7,$ $\vec r \cdot (2\hat i + 5\hat j + 3\hat k) = 9$ and through the point $(2,1,3)$. SA Q11 Find the equation of the plane through the line of intersection of the planes$x + y + z = 1$ and $2x + 3y + 4z = 5$ and which is perpendicular to the plane $x - y + z = 0$ . SA Q12 Find the angle between the planes whose vector equations are $\vec r \cdot (2\hat i + 2\hat j - 3\hat k) = 5$ and $\vec r \cdot (3\hat i - 3\hat j + 5\hat k) = 3.$ SA Q13 In the following cases, determine whether the given planes are parallel or perpendicular and in case they are neither, find the angles between them.(a) $7x + 5y + 6z + 30 = 0$ and $3x - y - 10z + 4 = 0$
(b) $2x + y + 3z - 2 = 0$ and $x - 2y + 5 = 0$
(c) $2x - 2y + 4z + 5 = 0$ and $3x - 3y + 6z - 1 = 0$
(d) $2x - y + 3z - 1 = 0$ and $2x - y + 3z + 3 = 0$
(e) $4x + 8y + z - 8 = 0$ and $y + z - 4 = 0$
SA Q14 In the following cases, find the distance of each of the given points from the corresponding given plane.SA
Miscellaneous Exercise
Q1
Show that the line joining the origin to the point $(2,1,1)$ is perpendicular to the line determined by the points $(3,5, - 1),(4,3, - 1).$
SA
Q2
If ${l_1},{m_1},{n_1}$ and ${l_2},{m_2},{n_2}$ are the direction cosines of two mutually perpendicular lines, then show that the direction cosines of the line perpendicular to both of these are ${m_1}{n_2} - {m_2}{n_1},{n_1}{l_2} - {n_2}{l_1},{l_1}{m_2} - {l_2}{m_1}.$
SA
Q3
Find the angle between the lines whose direction ratios are a, b, c and $b - c,c - a,a - b.$
SA
Q4
Find the equation of a line parallel to x-axis and passing through the origin.
SA
Q5
If the coordinates of the points $A,B,C,D$ be $(1,2,3)$ $(4,5,7),( - 4,3, - 6)$ and $(2,9,2)$respectively, then find the angle between the lines AB and CD.
SA
Q6
If the fines $\cfrac{{x - 1}}{{ - 3}} = \cfrac{{y - 2}}{{2k}} = \cfrac{{z - 3}}{2}$
and $\cfrac{{x - 1}}{{3k}} = \cfrac{{y - 1}}{1} = \cfrac{{z - 6}}{{ - 5}}$ are perpendicular, then find the value of k. SA Q7 Find the vector equation of the line passing through $(1,2,3)$ and perpendicular to the plane $\vec r \cdot (\hat i + 2\hat j - 5\hat k) + 9 = 0$. SA Q8 Find the equation of the plane passing through $(a,b,c)$ and parallel to the plane$\vec r \cdot (\hat i + \hat j + \hat k) = 2.$ SA Q9 Find the shortest distance between lines
$\vec r = (6\hat i + 2\hat j + 2\hat k + \lambda (\hat i - 2\hat j + 2\hat k)$ and
$\vec r = - 4\hat i - \hat k + \mu (3\hat i - 2\hat j - 2\hat k).$ SA Q10 Find the coordinates of the point where the line through $(5,1,6)$and $(3,4,1)$ crosses the XZ-plane. SA Q11 Find the coordinates of the point where the line through $(5,1,6)$ and $(3,4,1)$ crosses the ZX-plane. SA Q12 Find the coordinates of the point where the line through $(3, - 4, - 5)$ and $(2, - 3,1)$ crosses the plane$2x + y + z = 7$ . SA Q13 Find the equation of the plane passing through the point $( - 1,3,2)$ and perpendicular to each of the planes $x + 2y + 3z = 5$ and $3x + 3y + z = 0$. SA Q14 If the points $(1,1,p)$ and $( - 3,0,1)$ be equidistant from the plane $\vec r \cdot (3\hat i + 4\hat j - 12\hat k) + 13 = 0$ then find the value of p. SA Q15 Find the equation of the plane passing through the line of intersection of the planes $\vec r \cdot (\hat i + \hat j + \hat k) = 1$and $\vec r \cdot (2\hat i + 3\hat j - \hat k) + 4 = 0$and parallel to x-axis. SA Q16 If O be the origin and the coordinates of P be $(1,2, - 3)$, then, find the equation of the plane passing through P and perpendicular to O P. SA Q17 Find the equation of the plane which contains the line of intersection of the planes $\vec r \cdot (\hat i + 2\hat j + 3\hat k) - 4 = 0,\vec r \cdot (2\hat i + \hat j - \hat k) + 5 = 0$ and which is perpendicular to the plane $\vec r \cdot (5\hat i + 3\hat j - 6\hat k) + 8 = 0.$ SA Q18 Find the distance of the point $( - 1, - 5, - 10)$ from the point of intersection of the line $\vec r = 2\hat i - \hat j + 2\hat k + \lambda (3\hat i + 4\hat j + 2\hat k)$and the plane $\vec r \cdot (\hat i - \hat j + \hat k) = 5.$ SA Q19 Find the vector equation of the line passing through $(1,2,3)$ and parallel to the planes $\vec r \cdot (\hat i - \hat j + 2\hat k) = 5$ and $\vec r \cdot (3\hat i + \hat j + \hat k) = 6.$ SA Q20 Find the vector equation of the line passing through the point $(1,2, - 4)$ and perpendicular to the two lines:
$\cfrac{{x - 8}}{3} = \cfrac{{y + 19}}{{ - 16}} = \cfrac{{z - 10}}{7}$ and $\cfrac{{x - 15}}{3} = \cfrac{{y - 29}}{8} = \cfrac{{z - 5}}{{ - 5}}$ SA Q21 Prove that if a plane has the intercepts a,b, c and is at a distance o f p units from the origin, then $\cfrac{1}{{{a^2}}} + \cfrac{1}{{{b^2}}} + \cfrac{1}{{{c^2}}} = \cfrac{1}{{{p^2}}}.$ SA Q22 Distance between the two planes: $2x + 3y + 4z = 4$ and $4x + 6y + 8z = 12$ is
and $\cfrac{{x - 1}}{{3k}} = \cfrac{{y - 1}}{1} = \cfrac{{z - 6}}{{ - 5}}$ are perpendicular, then find the value of k. SA Q7 Find the vector equation of the line passing through $(1,2,3)$ and perpendicular to the plane $\vec r \cdot (\hat i + 2\hat j - 5\hat k) + 9 = 0$. SA Q8 Find the equation of the plane passing through $(a,b,c)$ and parallel to the plane$\vec r \cdot (\hat i + \hat j + \hat k) = 2.$ SA Q9 Find the shortest distance between lines
$\vec r = (6\hat i + 2\hat j + 2\hat k + \lambda (\hat i - 2\hat j + 2\hat k)$ and
$\vec r = - 4\hat i - \hat k + \mu (3\hat i - 2\hat j - 2\hat k).$ SA Q10 Find the coordinates of the point where the line through $(5,1,6)$and $(3,4,1)$ crosses the XZ-plane. SA Q11 Find the coordinates of the point where the line through $(5,1,6)$ and $(3,4,1)$ crosses the ZX-plane. SA Q12 Find the coordinates of the point where the line through $(3, - 4, - 5)$ and $(2, - 3,1)$ crosses the plane$2x + y + z = 7$ . SA Q13 Find the equation of the plane passing through the point $( - 1,3,2)$ and perpendicular to each of the planes $x + 2y + 3z = 5$ and $3x + 3y + z = 0$. SA Q14 If the points $(1,1,p)$ and $( - 3,0,1)$ be equidistant from the plane $\vec r \cdot (3\hat i + 4\hat j - 12\hat k) + 13 = 0$ then find the value of p. SA Q15 Find the equation of the plane passing through the line of intersection of the planes $\vec r \cdot (\hat i + \hat j + \hat k) = 1$and $\vec r \cdot (2\hat i + 3\hat j - \hat k) + 4 = 0$and parallel to x-axis. SA Q16 If O be the origin and the coordinates of P be $(1,2, - 3)$, then, find the equation of the plane passing through P and perpendicular to O P. SA Q17 Find the equation of the plane which contains the line of intersection of the planes $\vec r \cdot (\hat i + 2\hat j + 3\hat k) - 4 = 0,\vec r \cdot (2\hat i + \hat j - \hat k) + 5 = 0$ and which is perpendicular to the plane $\vec r \cdot (5\hat i + 3\hat j - 6\hat k) + 8 = 0.$ SA Q18 Find the distance of the point $( - 1, - 5, - 10)$ from the point of intersection of the line $\vec r = 2\hat i - \hat j + 2\hat k + \lambda (3\hat i + 4\hat j + 2\hat k)$and the plane $\vec r \cdot (\hat i - \hat j + \hat k) = 5.$ SA Q19 Find the vector equation of the line passing through $(1,2,3)$ and parallel to the planes $\vec r \cdot (\hat i - \hat j + 2\hat k) = 5$ and $\vec r \cdot (3\hat i + \hat j + \hat k) = 6.$ SA Q20 Find the vector equation of the line passing through the point $(1,2, - 4)$ and perpendicular to the two lines:
$\cfrac{{x - 8}}{3} = \cfrac{{y + 19}}{{ - 16}} = \cfrac{{z - 10}}{7}$ and $\cfrac{{x - 15}}{3} = \cfrac{{y - 29}}{8} = \cfrac{{z - 5}}{{ - 5}}$ SA Q21 Prove that if a plane has the intercepts a,b, c and is at a distance o f p units from the origin, then $\cfrac{1}{{{a^2}}} + \cfrac{1}{{{b^2}}} + \cfrac{1}{{{c^2}}} = \cfrac{1}{{{p^2}}}.$ SA Q22 Distance between the two planes: $2x + 3y + 4z = 4$ and $4x + 6y + 8z = 12$ is
(A) 2 units
(B) 4 units
(C) 8 units
(D) $\cfrac{2}{{\sqrt {29} }}$ units
SA Q23 The planes: $2x - y + 4z = 5$ and $5x - 2.5y + 10z = 6$ are(A) Perpendicular
(B) Parallel
(C) Intersect y-axis
(D) Pass through $\left( {0,0,\cfrac{5}{4}} \right)$
SA