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Vector Algebra and Analytic Geometry of Space

The Coordinates of a Point in Space

The Coordinate Representation of a Point

The coordinate representation of a point

The coordinates of a point PP in space are P(x0,y0,z0)P(x_0, y_0, z_0), where x0,y0,z0x_0, y_0, z_0 are the projections of the point PP onto the x,y,zx, y, z axes respectively.

The Distance between Two Points

The distance formula between two points

Let P1(x1,y1,z1)P_1(x_1, y_1, z_1) and P2(x2,y2,z2)P_2(x_2, y_2, z_2). Then the distance between them is:

d=(x2−x1)2+(y2−y1)2+(z2−z1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}

The Geometric Meaning

The distance between two points is the length of the line segment joining them, and it is the most basic distance measure in space.

Special Cases

  • when the two points lie on the same coordinate axis, the distance equals the absolute value of the difference of their coordinates
  • when the two points lie in the same coordinate plane, the formula reduces to the planar distance formula

Exercises

Exercise 1

Find the distance between the points A(1,2,3)A(1, 2, 3) and B(4,0,−1)B(4, 0, -1).

the distance between two pointscoordinate computationspatial geometry

Approach:

Use the distance formula between two points.

Steps:

  1. d=(4−1)2+(0−2)2+(−1−3)2d = \sqrt{(4-1)^2 + (0-2)^2 + (-1-3)^2}
  2. d=32+(−2)2+(−4)2d = \sqrt{3^2 + (-2)^2 + (-4)^2}
  3. d=9+4+16=29d = \sqrt{9 + 4 + 16} = \sqrt{29}

Answer: the distance is 29\sqrt{29}.

Exercise 2

Given the two points A(2,1,3)A(2, 1, 3) and B(4,3,1)B(4, 3, 1), find their distance and decide on which coordinate axis their projection is the farthest.

the distance between two pointscoordinate analysisspatial geometry

Approach:

Compute the distance and compare the distances along each coordinate axis.

Steps:

  1. The distance: d=(4−2)2+(3−1)2+(1−3)2=4+4+4=12=23d = \sqrt{(4-2)^2 + (3-1)^2 + (1-3)^2} = \sqrt{4 + 4 + 4} = \sqrt{12} = 2\sqrt{3}

  2. The distances along each axis:

    • the x-axis: |4-2| = 2
    • the y-axis: |3-1| = 2
    • the z-axis: |1-3| = 2

  3. The three distances are equal

Answer: the distance is 232\sqrt{3}, and the projection distances along the coordinate axes are all equal.

Summary

Symbols Used in This Article

SymbolTypePronunciation/ExplanationMeaning in This Article
P(x0,y0,z0)P(x_0, y_0, z_0)Point coordinatesPoint coordinatesThe coordinate representation of a point in space
ddDistancedistanceThe distance between two points
\sqrt{}Square rootsquare rootThe operation of taking a square root
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Analytic Geometry of Space
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Planes in Space

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