SolveItClass 11 · NCERT

NCERT Solutions · Class 11 Mathematics Introduction to Three Dimensional Geometry

13 questions · 13 still being checked

EXERCISE 11.1 1–4 (part 1 of 3)

  1. Exercise 1

    A point is on the x\displaystyle x-axis. What are its y\displaystyle y-coordinate and z\displaystyle z-coordinates?

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    NCERT’s answer
    $\displaystyle y$ and $\displaystyle z$ - coordinates are zero
    Reading coordinates off an axis.A point lies on the \(\displaystyle x\)-axis exactly when it can be reached from the origin by moving along the \(\displaystyle x\)-direction alone. Its displacement parallel to the \(\displaystyle y\)-axis and its displacement parallel to the \(\displaystyle z\)-axis are then both nil, and those displacements are what the \(\displaystyle y\)- and \(\displaystyle z\)-coordinates measure.So every point of the \(\displaystyle x\)-axis has the form \(\displaystyle (x,\,0,\,0)\).Both are zero: the \(\displaystyle y\)-coordinate is \(\displaystyle 0\) and the \(\displaystyle z\)-coordinate is \(\displaystyle 0\).

  2. Exercise 2

    A point is in the XZ-plane. What can you say about its y\displaystyle y-coordinate?

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    Reading coordinates off a coordinate plane.The XZ-plane is the plane determined by the \(\displaystyle x\)-axis and the \(\displaystyle z\)-axis. A point lies on it exactly when it needs no displacement in the \(\displaystyle y\)-direction to be reached from that plane — and the \(\displaystyle y\)-coordinate is precisely that displacement.So every point of the XZ-plane has the form \(\displaystyle (x,\,0,\,z)\).Its \(\displaystyle y\)-coordinate is \(\displaystyle 0\).

  3. Exercise 3

    Name the octants in which the following points lie: (1,2,3),(4,−2,3),(4,−2,−5),(4,2,−5),(−4,2,−5),(−4,2,5),(−3,−1,6)(−2,−4,−7).\begin{aligned} & (1,2,3),(4,-2,3),(4,-2,-5),(4,2,-5),(-4,2,-5),(-4,2,5), \\ & (-3,-1,6)(-2,-4,-7) . \end{aligned}

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    NCERT’s answer
    I, IV, VIII, V, VI, II, III, VII
    Sign pattern of the coordinates.The three coordinate planes cut space into eight octants, and the octant a point lies in is decided entirely by the signs of its three coordinates. The standard numbering is
    I \(\displaystyle (+,+,+)\), II \(\displaystyle (-,+,+)\), III \(\displaystyle (-,-,+)\), IV \(\displaystyle (+,-,+)\)
    V \(\displaystyle (+,+,-)\), VI \(\displaystyle (-,+,-)\), VII \(\displaystyle (-,-,-)\), VIII \(\displaystyle (+,-,-)\)
    (the first four octants sit above the XY-plane, the last four below it).Now read off the sign triple of each point:
    \(\displaystyle (1,2,3)\): \(\displaystyle (+,+,+)\) — octant I
    \(\displaystyle (4,-2,3)\): \(\displaystyle (+,-,+)\) — octant IV
    \(\displaystyle (4,-2,-5)\): \(\displaystyle (+,-,-)\) — octant VIII
    \(\displaystyle (4,2,-5)\): \(\displaystyle (+,+,-)\) — octant V
    \(\displaystyle (-4,2,-5)\): \(\displaystyle (-,+,-)\) — octant VI
    \(\displaystyle (-4,2,5)\): \(\displaystyle (-,+,+)\) — octant II
    \(\displaystyle (-3,-1,6)\): \(\displaystyle (-,-,+)\) — octant III
    \(\displaystyle (-2,-4,-7)\): \(\displaystyle (-,-,-)\) — octant VII
    The octants are I, IV, VIII, V, VI, II, III and VII respectively.

  4. Exercise 4

    Fill in the blanks:
    (i)
    The x\displaystyle x-axis and y\displaystyle y-axis taken together determine a plane known as ____\displaystyle \_\_\_\_.
    (ii)
    The coordinates of points in the XY-plane are of the form ____\displaystyle \_\_\_\_.
    (iii)
    Coordinate planes divide the space into ____\displaystyle \_\_\_\_ octants.

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    Definitions of the coordinate planes and octants.
    (i)
    Two intersecting axes determine a plane, and the plane is named after the pair. The \(\displaystyle x\)-axis and the \(\displaystyle y\)-axis therefore determine the XY-plane.
    (ii)
    A point of the XY-plane has no displacement in the \(\displaystyle z\)-direction, so its \(\displaystyle z\)-coordinate vanishes: the coordinates are of the form \(\displaystyle (x,\,y,\,0)\).
    (iii)
    The three coordinate planes each split space into two halves, and the three splits are independent, giving \(\displaystyle 2\times 2\times 2\) regions. So the coordinate planes divide space into eight octants.
    (i) XY-plane (ii) \(\displaystyle (x,y,0)\) (iii) eight.