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Three-Dimensional Geometry

Each subtopic includes About section, revision page link, 10 preview questions, and practice CTAs.

Direction Cosines and Ratios of a Line

Subtopic

Direction Cosines and Ratios of a Line under Three-Dimensional Geometry for Grade 12 ICSE.

About Topic & Revision

Preview questions (no answers)

  1. 1.

    What is the angle between two lines whose direction ratios are (1,0,0)(1, 0, 0) and (0,1,0)(0, 1, 0)?

    A.

    0∘0^\circ

    B.

    90∘90^\circ

    C.

    45∘45^\circ

    D.

    180∘180^\circ

  2. 2.

    If the direction ratios of a line are (1,1,0)(1, 1, 0), what is the angle the line makes with the zz-axis?

    A.

    90∘90^\circ

    B.

    45∘45^\circ

    C.

    0∘0^\circ

    D.

    60∘60^\circ

  3. 3.

    The direction ratios of the line segment joining (x1,y1,z1)(x_1, y_1, z_1) and the origin (0,0,0)(0, 0, 0) are:

    A.

    (0,0,0)(0, 0, 0)

    B.

    (1,1,1)(1, 1, 1)

    C.

    (x12,y12,z12)(\sqrt{x_1^2}, \sqrt{y_1^2}, \sqrt{z_1^2})

    D.

    (x1,y1,z1)(x_1, y_1, z_1)

Download the worksheet for Three-Dimensional Geometry - Direction Cosines and Ratios of a Line to practice offline. It includes additional chapter-level practice questions.

Equation of a Line (Vector and Cartesian)

Subtopic

Equation of a Line (Vector and Cartesian) under Three-Dimensional Geometry for Grade 12 ICSE.

About Topic & Revision

Preview questions (no answers)

  1. 1.

    The Cartesian equation of a line passing through the point (5,2,−4)(5, 2, -4) and having direction ratios (3,7,−2)(3, 7, -2) is:

    A.

    x+53=y+27=z−4−2\frac{x+5}{3} = \frac{y+2}{7} = \frac{z-4}{-2}

    B.

    x−35=y−72=z+2−4\frac{x-3}{5} = \frac{y-7}{2} = \frac{z+2}{-4}

    C.

    x−53=y−27=z−4−2\frac{x-5}{3} = \frac{y-2}{7} = \frac{z-4}{-2}

    D.

    x−53=y−27=z+4−2\frac{x-5}{3} = \frac{y-2}{7} = \frac{z+4}{-2}

  2. 2.

    Which of the following represents the vector equation of a line passing through the point (2,3,4)(2, 3, 4) and parallel to the zz-axis?

    A.

    r⃗=(2i^+3j^+4k^)+λ(i^+j^+k^)\vec{r} = (2\hat{i} + 3\hat{j} + 4\hat{k}) + \lambda(\hat{i} + \hat{j} + \hat{k})

    B.

    r⃗=λ(2i^+3j^+4k^)\vec{r} = \lambda(2\hat{i} + 3\hat{j} + 4\hat{k})

    C.

    r⃗=(2i^+3j^+4k^)+λk^\vec{r} = (2\hat{i} + 3\hat{j} + 4\hat{k}) + \lambda\hat{k}

    D.

    r⃗=k^+λ(2i^+3j^+4k^)\vec{r} = \hat{k} + \lambda(2\hat{i} + 3\hat{j} + 4\hat{k})

  3. 3.

    The direction ratios of the line passing through the points P(1,−1,3)P(1, -1, 3) and Q(2,4,−1)Q(2, 4, -1) are:

    A.

    (1,5,−4)(1, 5, -4)

    B.

    (3,3,2)(3, 3, 2)

    C.

    (1,3,−2)(1, 3, -2)

    D.

    (1,5,4)(1, 5, 4)

Download the worksheet for Three-Dimensional Geometry - Equation of a Line (Vector and Cartesian) to practice offline. It includes additional chapter-level practice questions.

Shortest Distance between Skew Lines

Subtopic

Shortest Distance between Skew Lines under Three-Dimensional Geometry for Grade 12 ICSE.

About Topic & Revision

Preview questions (no answers)

  1. 1.

    If the shortest distance between two lines is zero, what can be concluded about the lines?

    A.

    They are parallel

    B.

    They are skew

    C.

    They are coplanar and intersect

    D.

    They are in perpendicular planes

  2. 2.

    The shortest distance between the lines x=y,z=2x=y, z=2 and x=−y,z=4x=-y, z=4 is:

    A.

    1 unit

    B.

    4 units

    C.

    2 units

    D.

    0 units

  3. 3.

    Find the shortest distance between the lines r⃗=(1,1,1)+λ(1,0,0)\vec{r} = (1, 1, 1) + \lambda(1, 0, 0) and r⃗=(2,3,4)+μ(0,0,1)\vec{r} = (2, 3, 4) + \mu(0, 0, 1).

    A.

    1 unit

    B.

    2 units

    C.

    3 units

    D.

    0 units

Download the worksheet for Three-Dimensional Geometry - Shortest Distance between Skew Lines to practice offline. It includes additional chapter-level practice questions.

Equation of a Plane (Vector and Cartesian)

Subtopic

Equation of a Plane (Vector and Cartesian) under Three-Dimensional Geometry for Grade 12 ICSE.

About Topic & Revision

Preview questions (no answers)

  1. 1.

    If a plane passes through (2,0,0)(2, 0, 0), (0,3,0)(0, 3, 0) and (0,0,4)(0, 0, 4), its equation is:

    A.

    x2+y3+z4=1\frac{x}{2} + \frac{y}{3} + \frac{z}{4} = 1

    B.

    2x+3y+4z=122x + 3y + 4z = 12

    C.

    x+y+z=9x + y + z = 9

    D.

    x2+y3+z4=0\frac{x}{2} + \frac{y}{3} + \frac{z}{4} = 0

  2. 2.

    The equation of the plane x=0x = 0 represents:

    A.

    The XYXY-plane

    B.

    The XZXZ-plane

    C.

    The YZYZ-plane

    D.

    A plane parallel to the XX-axis

  3. 3.

    Which plane is parallel to the ZZ-axis?

    A.

    x+y=2x + y = 2

    B.

    z=5z = 5

    C.

    x+y+z=0x + y + z = 0

    D.

    z=xz = x

Download the worksheet for Three-Dimensional Geometry - Equation of a Plane (Vector and Cartesian) to practice offline. It includes additional chapter-level practice questions.

Angle between Lines, Planes, and a Line and a Plane

Subtopic

Angle between Lines, Planes, and a Line and a Plane under Three-Dimensional Geometry for Grade 12 ICSE.

About Topic & Revision

Preview questions (no answers)

  1. 1.

    If the angle between the line x−11=y−11=z−1n\frac{x-1}{1} = \frac{y-1}{1} = \frac{z-1}{n} and the plane x+y+z=5x+y+z=5 is 30∘30^\circ, find nn. (Simple case check: n=1n=1 gives sin⁡θ=3/3=1\sin \theta = 3/3 = 1)

    A.

    11

    B.

    00

    C.

    22

    D.

    −1-1

  2. 2.

    Angle between the lines x=yx=y and x=−yx=-y in the xyxy-plane (viewed in 3D) is:

    A.

    0∘0^\circ

    B.

    45∘45^\circ

    C.

    90∘90^\circ

    D.

    60∘60^\circ

  3. 3.

    Find the angle between the line x−11=y1=z−11\frac{x-1}{1} = \frac{y}{1} = \frac{z-1}{1} and the plane x+y+z=10x+y+z=10.

    A.

    0∘0^\circ

    B.

    30∘30^\circ

    C.

    60∘60^\circ

    D.

    90∘90^\circ

Download the worksheet for Three-Dimensional Geometry - Angle between Lines, Planes, and a Line and a Plane to practice offline. It includes additional chapter-level practice questions.

Distance of a Point from a Plane

Subtopic

Distance of a Point from a Plane under Three-Dimensional Geometry for Grade 12 ICSE.

About Topic & Revision

Preview questions (no answers)

  1. 1.

    Find the distance of the point (1,1,1)(1, 1, 1) from the plane 3x+4y+12z+13=03x + 4y + 12z + 13 = 0.

    A.

    3213\frac{32}{13}

    B.

    2

    C.

    3013\frac{30}{13}

    D.

    1913\frac{19}{13}

  2. 2.

    What is the distance from the point (2,2,0)(2, 2, 0) to the plane x+y+z=1x + y + z = 1?

    A.

    3\sqrt{3}

    B.

    3

    C.

    13\frac{1}{\sqrt{3}}

    D.

    333\sqrt{3}

  3. 3.

    Calculate the distance from (0,1,0)(0, 1, 0) to the plane x+y+z=4x + y + z = 4.

    A.

    3\sqrt{3}

    B.

    33\frac{3}{\sqrt{3}}

    C.

    3

    D.

    1

Download the worksheet for Three-Dimensional Geometry - Distance of a Point from a Plane to practice offline. It includes additional chapter-level practice questions.