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Home Vectors Vectors and Plane Geometry Examples Example 5: Points On a Line (Or Not)
 
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Example 5: Midpoint of a Linesegment (Proof)

Let A and B be two distinct points. Prove that the midpoint of the line segment AB is the point P with position vector P = ½A + ½B.

10_vectors-94.gif

Figure 10.2.11

PROOF

We shall prove that the point P is on the line AB and is equidistant from A and B (see Figure 10.2.11). The line through A and B has the direction vector D = B - A. The vector P has the form

P = ½A + ½B = A + ½(B - A) = A + ½D.

Therefore by Theorem 1, P is on the line AB. To prove that P is equidistant, we show that the vector from A to P is the same as the vector from P to B

P - A = ½A + ½B - A = ½B - ½A.

B - P = B - ½A - ½B = ½B - ½A.
Home Vectors Vectors and Plane Geometry Examples Example 5: Points On a Line (Or Not)

Last Update: 2006-11-15