CBSE Class 12 Mathematics Three Dimensional Geometry VBQs Set 03

Short Answer Questions–I

Question. Find the coordinates of the point where the line \( \frac{x+1}{2} = \frac{y+2}{3} = \frac{z+3}{4} \) meets the plane \( x + y + 4z = 6 \).
Answer: (1, 1, 1)

Question. If the line drawn from the point (–2, –1, –3) meets a plane at right angle at the point (1, –3, 3), find the equation of the plane.
Answer: \( 3x - 2y + 6z - 27 = 0 \)

Question. Find the distance of the point whose position vector is \( (2\hat{i} + \hat{j} - \hat{k}) \) from the plane \( \vec{r} \cdot (\hat{i} - 2\hat{j} + 4\hat{k}) = 9 \).
Answer: \( \frac{13}{\sqrt{21}} \)

Question. Write the unit vector normal to the plane \( x + 2y + 3z - 6 = 0 \).
Answer: \( \frac{1}{\sqrt{14}}\hat{i} + \frac{2}{\sqrt{14}}\hat{j} + \frac{3}{\sqrt{14}}\hat{k} \)

Question. Find the vector equation of a plane which is at a distance of 5 units from the origin and its normal vector is \( 2\hat{i} - 3\hat{j} + 6\hat{k} \).
Answer: \( \vec{r} \cdot \left( \frac{2}{7}\hat{i} - \frac{3}{7}\hat{j} + \frac{6}{7}\hat{k} \right) = 5 \)

Question. Find the equation of the plane passing through the line of intersection of the planes \( 2x + 2y - 3z = 7 \) and \( 2x + 5y + 3z = 9 \) the point (2, 1, 3).
Answer: \( 38x + 68y + 3z = 153 \)

Question. Find the angle between the planes, whose vector equations are \( \vec{r} \cdot (2\hat{i} + 2\hat{j} - 3\hat{k}) = 5 \) and \( \vec{r} \cdot (3\hat{i} - 3\hat{j} + 5\hat{k}) = 3 \).
Answer: \( \cos^{-1}\left( \frac{15}{\sqrt{731}} \right) \)

Short Answer Questions–II

Question. Find the shortest distance between the following pair of skew lines: \( \frac{x-1}{2} = \frac{2-y}{3} = \frac{z+1}{4} , \frac{x+2}{1} = \frac{y-3}{2} = \frac{z}{3} \).
Answer: \( \frac{42}{\sqrt{390}} \) units

Question. Show that the lines \( \frac{x-1}{3} = \frac{y-1}{-1} = \frac{z+1}{0} \) and \( \frac{x-4}{2} = \frac{y}{0} = \frac{z+1}{3} \) intersect. Find their point of intersection.
Answer: (4, 0, –1)

Question. Find the coordinates of the foot of perpendicular drawn from the point A(–1, 8, 4) to the line joining the points \( B(0, -1, 3) \) and \( C(2, -3, -1) \). Hence find the image of the point A in the line BC.
Answer: (–2, 1, 7); (–3, –6, 10)

Question. Find the equation of plane passing through the points \( A(3, 2, 1), B(4, 2, -2) \) and \( C(6, 5, -1) \) and hence find the value of \(\lambda\) for which \( A(3, 2, 1), B(4, 2, -2), C(6, 5, -1) \) and \( D(\lambda, 5, 5) \) are coplanar.
Answer: \( 9x - 7y + 3z - 16 = 0 \); \( \lambda = 4 \)

Question. Prove that the line through \( A(0, -1, -1) \) and \( B(4, 5, 1) \) intersects the line through \( C(3, 9, 4) \) and \( D(-4, 4, 4) \).
Answer: The lines intersect at (2, 2, 0).

Question. Show that the following two lines are coplanar: \( \frac{x-a+d}{\alpha-\delta} = \frac{y-a}{\alpha} = \frac{z-a-d}{\alpha+\delta} \) and \( \frac{x-b+c}{\beta-\gamma} = \frac{y-b}{\beta} = \frac{z-b-c}{\beta+\gamma} \).
Answer: The lines are coplanar since the determinant condition for coplanarity is satisfied.

Question. Find the acute angle between the plane \( 5x - 4y + 7z - 13 = 0 \) and the y-axis.
Answer: \( \sin^{-1} \left( \frac{4}{3\sqrt{10}} \right) \)

Question. Find the equation of a plane which is at a distance of \( 3\sqrt{3} \) units from origin and the normal to which is equally inclined to the coordinate axes.
Answer: \( x + y + z = 9 \)

Question. Let \( P(3, 2, 6) \) be a point in the space and Q be a point on the line \( \vec{r} = (\hat{i} - \hat{j} + 2\hat{k}) + \mu(-3\hat{i} + \hat{j} + 5\hat{k}) \), then find the value of \(\mu\) for which the vector \( \vec{PQ} \) is parallel to the plane \( x - 4y + 3z = 1 \).
Answer: \( \mu = \frac{1}{4} \)

Question. Find the vector and cartesian equations of the plane which bisects the line joining the points (3, –2, 1) and (1, 4, –3) at right angles.
Answer: \( \vec{r} \cdot (\hat{i} - 3\hat{j} + 2\hat{k}) + 3 = 0 \); \( x - 3y + 2z + 3 = 0 \)

Question. Find the distance of the point \( P(3, 4, 4) \) from the point, where the line joining the points \( A(3, -4, -5) \) and \( B(2, -3, 1) \) intersect the plane \( 2x + y + z = 7 \).
Answer: 7 units

Question. Find the distance of the point (1, –2, 3) from the plane \( x - y + z = 5 \) measured parallel to the line whose direction cosines are proportional to 2, 3, –6.
Answer: 1 unit

Question. Find the equation of the plane containing two parallel lines \( \frac{x-1}{2} = \frac{y+1}{-1} = \frac{z}{3} \) and \( \frac{x}{4} = \frac{y-2}{-2} = \frac{z+1}{6} \). Also, find if the plane thus obtained contains the line \( \frac{x-2}{3} = \frac{y-1}{1} = \frac{z-2}{5} \) or not.
Answer: \( 8x + y - 5z = 7 \); yes the plane contains the given line.

Question. Find the vector and cartesian equations of a line through the point (1, –1, 1) and perpendicular to the lines joining the points (4, 3, 2), (1, –1, 0) and (1, 2, –1), (2, 1, 1).
Answer: \( \frac{x-1}{10} = \frac{y+1}{-4} = \frac{z-1}{-7} \); \( \vec{r} = (\hat{i} - \hat{j} + \hat{k}) + \lambda(10\hat{i} - 4\hat{j} - 7\hat{k}) \)

Question. 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 \( \vec{r} = (s+1)\hat{i} + (2s-1)\hat{j} + (2s+1)\hat{k} \).
Answer: \( \frac{8}{\sqrt{29}} \) units

Question. Find the vector and cartesian equations of the line passing through the point P (1, 2, 3) and parallel to the planes \( \vec{r} \cdot (\hat{i} - \hat{j} + 2\hat{k}) = 5 \) and \( \vec{r} \cdot (3\hat{i} + \hat{j} + \hat{k}) = 6 \).
Answer: \( \vec{r} = (\hat{i} + \hat{j} + \hat{k}) + \lambda(-3\hat{i} + 5\hat{j} + 4\hat{k}) \); \( \frac{x-1}{-3} = \frac{y-2}{5} = \frac{z-3}{4} \)

Long Answer Questions

Question. Find the equation of the plane passing through the point P(1, 1, 1) and containing the line \( \vec{r} = (-3\hat{i} + \hat{j} + 5\hat{k}) + \lambda(3\hat{i} - \hat{j} - 5\hat{k}) \). Also, show that the plane contains the line \( \vec{r} = (-\hat{i} + 2\hat{j} + 5\hat{k}) + \mu(\hat{i} - 2\hat{j} - 5\hat{k}) \).
Answer: \( \vec{r} \cdot (\hat{i} - \hat{j} + \hat{k}) = 0 \)

Question. Find the vector and cartesian equations of the plane passing through the intersection of the planes \( \vec{r} \cdot (\hat{i} + \hat{j} + \hat{k}) = 6 \) and \( \vec{r} \cdot (2\hat{i} + 3\hat{j} + 4\hat{k}) = -5 \) and the point (1, 1, 1).
Answer: \( 20x + 23y + 26z - 69 = 0 \); \( \vec{r} \cdot (20\hat{i} + 23\hat{j} + 26\hat{k}) = 69 \)

Question. Find the value of k for which the following lines are perpendicular to each other: \( \frac{x+3}{k-5} = \frac{y-1}{1} = \frac{5-z}{-2k-1} \) and \( \frac{x+2}{-1} = \frac{2-y}{-k} = \frac{z}{5} \). Hence, find the equation of the plane containing the above lines.
Answer: \( k = -1 \); \( 4x + 31y + 7z = 54 \)

Question. Show that the lines: \( \vec{r} = \hat{i} + \hat{j} + \hat{k} + \lambda(\hat{i} - \hat{j} + \hat{k}) \) and \( \vec{r} = 4\hat{j} + 2\hat{k} + \mu(2\hat{i} - \hat{j} + 3\hat{k}) \) are coplanar. Also, find the equation of the plane containing these lines.
Answer: \( \vec{r} \cdot (-2\hat{i} - \hat{j} + \hat{k}) + 2 = 0 \)

Question. Show that the lines \( \vec{r} = (-3\hat{i} + \hat{j} + 5\hat{k}) + \lambda(3\hat{i} - \hat{j} - 5\hat{k}) \) and \( \vec{r} = (-\hat{i} + 2\hat{j} + 5\hat{k}) + \mu(\hat{i} - 2\hat{j} - 5\hat{k}) \) are co-planar. Also, find the equation of the plane containing these lines.
Answer: \( x - 2y + z = 0 \)

Question. Find the position vector of the foot of perpendicular and the perpendicular distance from the point P with position vector \( 2\hat{i} + 3\hat{j} + 4\hat{k} \) to the plane \( \vec{r} \cdot (2\hat{i} + \hat{j} + 3\hat{k}) - 26 = 0 \). Also, find image of P in the plane.
Answer: \( 3\hat{i} + \frac{7}{2}\hat{j} + \frac{11}{2}\hat{k} \); \( \sqrt{\frac{7}{2}} \) units; (4, 4, 7)

Question. Find the coordinates of the foot of perpendicular and perpendicular distance from the point P(4, 3, 2) to the plane \( x + 2y + 3z = 2 \). Also find the image of P in the plane.
Answer: (3, 1, –1); \( \sqrt{14} \) sq. units; (2, –1, –4)

Question. Find the equation of the plane which contains the line of intersection of the planes \( x + 2y + 3z - 4 = 0 \) and \( 2x + y - z + 5 = 0 \) and whose x-intercept is twice its z-intercept. Hence write the vector equation of a plane passing through the point (2, 3, –1) and parallel to the plane obtained above.
Answer: \( 13x + 14y + 11z = 0 \) or \( 7x + 11y + 14z - 15 = 0 \); \( \{ \vec{r} - (2\hat{i} + 3\hat{j} - \hat{k}) \} \cdot (7\hat{i} + 11\hat{j} + 14\hat{k}) = 0 \)

Question. Find the coordinate of the point where the line through (3, –4, –5) and (2, –3, 1) crosses the plane, passing through the points (2, 2, 1), (3, 0, 1) and (4, –1, 0).
Answer: (1, –2, 7)

Question. Show that lines \( \vec{r} = (\hat{i} + \hat{j} - \hat{k}) + \lambda(3\hat{i} - \hat{j}) \) and \( \vec{r} = (4\hat{i} - \hat{k}) + \mu(2\hat{i} + 3\hat{k}) \) intersect. Also find their point of intersection.
Answer: (4, 0, –1)

Question. Find the distance between the point (7, 2, 4) and the plane determine by the points \( A(2, 5, -3), B(-2, -3, 5) \) and \( C(5, 3, -3) \).
Answer: 29 units

Question. Show that the lines \( \frac{x-2}{1} = \frac{y-2}{3} = \frac{z-3}{1} \) and \( \frac{x-2}{1} = \frac{y-3}{4} = \frac{z-4}{2} \) intersect. Also, find the co-ordinate of the point of intersection and equation of the plane containing the two lines.
Answer: (1, –1, 2); \( 2x - y + z - 5 = 0 \)