Finding the Vector Equation of a Line Passing Through Two Given Points

The general equation of a line passing through two pointsandis given by whereandcan be in two or three dimension ieort is a parameter – the value ofdetermines a unique point on the line. Any point on the line can be found by substituting a suitable value ofThe pointsandmay be expressed in two different forms, but both forms label the same point in space:

is equivalent to the vector

is equivalent to the column vector

Example: Find the equation of the line passing through the the pointsand

If instead ofcomponent form we used column vector form we word have


The two forms are equivalent, which can be seen from (1) by writing

So that

Conversely we can change from column vector form to component form:

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