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Introduction To Three Dimensional Geometry

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Introduction To Three Dimensional Geometry

The coordinates of points in the XY-plane are of the form __________.

(x, y, 0).

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The x-axis and y-axis taken together determine a plane known as ____________.

XOY or the XY-plane.

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Coordinate planes divide the space into _______ octants.

eight

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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.