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Introduction To Three Dimensional Geometry

Long Answer Type

21.

The co-ordinates of a point H are (-2, 3, -4). Find
(i) The co-ordinates of the foot of perpendicular from this point to ZX-plane.
(ii) The co-ordinates of the foot of perpendicular from this point to the z-axis.
(iii) The length of perpendicular from this point to XY-plane.

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Fill In the Blanks

22. The co-ordinates of the foot of perpendicular from point P (a, b, c) to the x-axis are _________.

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23. The co-ordinates of the foot of perpendicular from point (2, -3, 4) to ZX-plane are ____________.

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24. The length of perpendicular from point P (-2, -3, -5) to XY-plane is ____________.

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25. The co-ordinates of a point (a, b, c) on z-axis are of the type __________.

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26. The co-ordinates of a point (a, b, c) on YZ-plane are of the type _________.

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Short Answer Type

27. The co-ordinates of a point P are (x, y, z), where x, y, z > 0. If the length of perpendicular from it to ZX-plane is ¹/₂ its perpendicular distance from YZ-plane, then find a relation between x and y.

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28. Find the distance between the following pairs of points:
(2, -1, 3) and (-2, -1, 3)

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29. Find the distance between the following pairs of points:
(-1, 3, -4) and (1, -3, 4)

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30. Show that the points (-2, 3, 5), (1, 2, 3) and (7, 0, -1) are collinear.

Given points are : A(-2, 3, 5), B(1, 2, 3) and C(7, 0, -1)
Using distance formula, $space space space space space AB equals square root of left parenthesis straight x subscript 1 minus straight x subscript 2 right parenthesis squared plus left parenthesis straight y subscript 1 minus straight y subscript 2 right parenthesis squared plus left parenthesis straight z subscript 1 minus straight z subscript 2 right parenthesis squared end root$ , we have
$AB equals square root of left parenthesis negative 2 minus 1 right parenthesis squared plus left parenthesis 3 minus 2 right parenthesis squared plus left parenthesis 5 minus 3 right parenthesis squared end root equals square root of left parenthesis negative 3 right parenthesis squared plus left parenthesis 1 right parenthesis squared plus left parenthesis 2 right parenthesis squared end root$
$equals square root of 9 plus 1 plus 4 end root equals square root of 14$

$BC equals square root of left parenthesis 1 minus 7 right parenthesis squared plus left parenthesis 2 minus 0 right parenthesis squared plus left parenthesis 3 plus 1 right parenthesis squared end root space equals space square root of left parenthesis negative 6 right parenthesis squared plus left parenthesis 2 right parenthesis squared plus left parenthesis 4 right parenthesis squared end root$
$equals space space square root of 36 plus 4 plus 16 end root space equals space square root of 56 space equals space 2 square root of 14$

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$equals space square root of 81 plus 9 plus 36 end root space equals space square root of 126 space equals space 3 square root of 14$
Now, AB + BC = $square root of 14 plus 2 square root of 14 equals 3 square root of 14$ = CA
Hence, points A, B, C are collinear.

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