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    Assignment No 8 (3-D Geometry)

    Uploaded by

    Ranu Games
    This document contains 15 multi-part math problems involving vectors and 3D geometry. The problems involve finding lengths of vectors, distances between points, properties of triangles, cubes, tetrahedrons and other 3D shapes, projections, planes, lines of intersection, volumes, and shortest distances. The final problem asks to show a line intersects two planes in the same point, meaning the line is coplanar with the line of intersection of the planes.

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    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd

    Assignment No 8 (3-D Geometry)

    Uploaded by

    Ranu Games
    153 views2 pages
    This document contains 15 multi-part math problems involving vectors and 3D geometry. The problems involve finding lengths of vectors, distances between points, properties of triangles, cubes, tetrahedrons and other 3D shapes, projections, planes, lines of intersection, volumes, and shortest distances. The final problem asks to show a line intersects two planes in the same point, meaning the line is coplanar with the line of intersection of the planes.

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    Assignment No 8(3-D Geometry)

    Original Title

    Assignment No 8(3-D Geometry)

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    This document contains 15 multi-part math problems involving vectors and 3D geometry. The problems involve finding lengths of vectors, distances between points, properties of triangles, cubes, tetrahedrons and other 3D shapes, projections, planes, lines of intersection, volumes, and shortest distances. The final problem asks to show a line intersects two planes in the same point, meaning the line is coplanar with the line of intersection of the planes.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    153 views2 pages

    Assignment No 8 (3-D Geometry)

    Uploaded by

    Ranu Games
    This document contains 15 multi-part math problems involving vectors and 3D geometry. The problems involve finding lengths of vectors, distances between points, properties of triangles, cubes, tetrahedrons and other 3D shapes, projections, planes, lines of intersection, volumes, and shortest distances. The final problem asks to show a line intersects two planes in the same point, meaning the line is coplanar with the line of intersection of the planes.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    of 2

    Assignment No 8 (3-D Geometry)

    1. For each of the following vectors, find its length and find a vector of length one (unit" vector)
    parallel to it.
    a = (4;-4; 2); b = (2; 1; 0) ; c = (4; 0; 1;-2; 0)

    2. Find the distances between each of the following pairs of points


    (a) (8;-4; 2), (-6; 1; 0); (b) (1; 1; 1), (5;-7;-7);
    (c) (3; 0; 1; 4), (-2; 6; 1; 3);

    3.A triangle has vertices A,B and C which have co-ordinate vectors (4; 1; 7); (7;-4; 6) and (6; 2; 8)
    respectively. Find the lengths of the sides of the triangle and deduce that the triangle is right-angled.

    4. A cube has vertices at the 8 points O (0; 0; 0), A(1; 0; 0), B (1; 1; 0), C (0; 1; 0), D(0; 0; 1),
    E (1; 0; 1), F (1; 1; 1), G(0; 1; 1). Sketch the cube, and then find the angle between the diagonals
    OF and AG .

    4.Find the following projections:


    (a) the projection of (2; 1; 4) on (1;-2; 1);
    (b) the projection of (2;-1; 2) on (-1; 3; 0);
    (c) the projection of (-2; 2; 7) on the direction of the line x = (1; 0; 2) + k(-1; 1; 2).

    5. Find the areas of, and the normals to the planes of, the following parallelograms:
    (a) the parallelogram spanned by (1; 3; 2) and (0; 2; 4);
    (b) a parallelogram which has vertices at the three points A(0; 2; 1), B(-1; 3; 0) and C(3; 1; 2).

    6. Find the areas of the triangles with the following vertices:


    (a) A(0; 2; 1), B(-1; 3; 0) and C(3; 1; 2);
    (b) A(2; 2; 0), B(-1; 0; 2) and C(0; 4; 3).

    7. A tetrahedron has vertices A, B, C and D with coordinate vectors for the points being a =(0; 1; 2),
    b = (-1; 4; 1), c = (1; 0; 3) and d = (-3; 1; 2). Find parametric vector equations for the two altitudes
    of the tetrahedron which pass through the vertices A and B, and determine whether the two altitudes
    intersect or not.
    Note: An altitude of a tetrahedron through a vertex is a line through the vertex and perpendicular to
    the opposite face.

    8. Find the volumes of the following parallelepipeds:


    (a) the parallelepiped spanned by (2; 1; 3), (4; 1; 2) and (0; 2; 1);
    (b) a parallelepiped which has vertices at the four points A(2; 1; 3), B(-2; 1; 4), C(0; 4; 1) and
    D(3;-1; 0).

    9. Find parametric vector, point-normal, and Cartesian forms for the following planes:
    (a) the plane through (1; 2;-2) perpendicular to (-1; 1; 2);
    (b) the plane through (1; 2;-2) parallel to (-1; 1; 2) and (2; 3; 1);
    (d) the plane through the three points (1; 2;-2), (-1; 1; 2) and (2; 3; 1);
    (d) the plane with intercepts -1, 2 and -4 on the x , y and z axes;
    10.
    (a) Find the plane through (1; 2;-2) which is parallel to (-1; 2;-2)

    (b) Find the line of intersection of the planes x .(1; 2; 3) = 0 and x = λ 1 (2,1,2) + λ 2 (1,0,1)

    ( c) Find the following projections:


    (i) the projection of (2; 3; 8) on the normal to the plane 2x+2y+z 4
    (ii) the projection of (2; 2;-1) on the line of intersection of the planes x ⋅ (1,−1,3) = 0 and
    x = λ 1 (2,1,2) + λ 2 (3,1,−3)

    11. Find the shortest distances between


    (a) the point (-2; 1; 5) and the line x = (1; 2;-5) + k(6; 3;-4);
    x −1 y−2 z −3
    (b) the point (0; 3; 8) and the line = =
    1 −1 4
    (c) the point (11; 2; 1) and the line of intersection of the planes
    x ⋅ (1,−1,3) = 0 and x = λ 1 (2,1,2) + λ 2 (3,1,−3)

    12. Find the shortest distances between


    (a) the point (2, 6,-5) and the plane ( x − (1, 2, 3)) ⋅ (−2, 4, 4) = 0
    (b) the point (1, 4, 1) and the plane 2x-2y+y = 5
    (c) the point (1, 2, 1) and the plane with intercepts at 3,-1,2 on the three axes;
    (d) the origin and the plane through the three points (2, 1,3), (5,3,1) and (5,1,2).

    13. Find the shortest distances between


    (a) the line through (1,2, 3) parallel to (3 , 0, 1) and the line through the point (0, 2, 5) parallel to
    (3,-2, 2);
    (b) the line through the points (1, 3, 1) and (1, 5, -1) and the line through the points (0, 2, 1) and
    (1, 2, -3),
    x −1 y−2 z −3
    (c) the lines x = (2, 7, 8) + k(4; 3;-5) and = =
    − 10 1 4

    14. Four points have coordinate vectors A(1; 0; 2), B(-1; 2; 3), C(0; 1; 1) and D(2; 0;-1).
    (a) Find the parametric vector equations of the line through A and B and the line through C and D.
    (b) Find the shortest distance between the lines AB and CD.
    (c) Find the point P on AB and point Q on CD such that PQ is the shortest distance between the
    lines AB and CD.

    x−7 y+3 z−4


    15. Show that the line = = intersects the planes 6x + 4y – 5z = 4 and
    2 −1 1
    x – 5y + 2z = 12 in the same point and deduce that the line is coplanar with the line of intersection
    of the planes

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