Equation of a line is defined as y= mx+c, where c is the y-intercept and m is the slope. Vectors can be defined as a quantity possessing both direction and magnitude. Position vectors simply denote the position or location of a point in the three-dimensional Cartesian system with respect to a reference origin. Further, we shall study in detail about vectors and Cartesian equation of a line in three dimensions. It is known that we can uniquely determine a line if:

• It passes through a particular point in a specific direction, or
• It passes through two unique points

Let us study each case separately and try to determine the equation of a line in both the given cases.

Equation of a Line passing through a point and parallel to a vector

Let us consider that the position vector of the given point be  \(\vec{a} \) with respect to the origin. The line passing through point A is given by l and it is parallel to the vector \(\vec{k} \) as shown below. Let us choose any random point R on the line l and its position vector with respect to origin of the rectangular co-ordinate system is given by \(\vec{r} \).

Since the line segment, \(\overline{AR} \) is parallel to vector  \(\vec{k} \), therefore for any real number α,

\(\overline{AR} \) = α \(\vec{k} \)

Also, \(\overline{AR} \)=\(\overline{OR} \) – \(\overline{OA} \)

Therefore, α \(\vec{r} \) = \(\vec{r} \) – \(\vec{a} \)

From the above equation it can be seen that for different values of α, the above equations give the position of any arbitrary point R lying on the line passing through point A and parallel to vector k. Therefore, the vector equation of a line passing through a given point and parallel to a given vector is given by:

\(\vec{r} \) = \(\vec{a} \) + α\(\vec{k} \)

If the three-dimensional co-ordinates of the point ‘A’ are given as (x₁, y₁, z₁) and the direction cosines of this point is given as a, b, c then considering the rectangular co-ordinates of point R as (x, y, z):

Substituting these values in the vector equation of a line passing through a given point and parallel to a given vector and equating the coefficients of unit vectors i, j and k, we have,

<

Eliminating α we have:

This gives us the Cartesian equation of line.

Equation of a Line passing through two given points

Let us consider that the position vector of the two given points A and B be \(\vec{a} \) and \(\vec{b} \) with respect to the origin. Let us choose any random point R on the line and its position vector with respect to origin of the rectangular co-ordinate system is given by \(\vec{r} \).

Point R lies on the line AB if and only if the vectors  \(\overline{AR} \) and \(\overline{AB} \) are collinear. Also,

\(\overline{AR} \) = \(\vec{r} \) – \(\vec{a}\)

\(\overline{AB} \) = \(\vec{b} \) – \(\vec{a}\)

Thus R lies on AB only if;
\(\vec{r} – \vec{a} = \alpha (\vec{b} – \vec{a})\)

Here α is any real number.
From the above equation it can be seen that for different values of α, the above equation gives the position of any arbitrary point R lying on the line passing through point A and B. Therefore, the vector equation of a line passing through two given points is given by:
\(\vec{r} = \vec{a} + \alpha (\vec{b} – \vec{a})\)

If the three-dimensional coordinates of the points A and B are given as (x₁, y₁, z₁) and (x₂, y₂, z₂) then considering the rectangular co-ordinates of point R as (x, y, z)

Substituting these values in the vector equation of a line passing through two given points and equating the coefficients of unit vectors i, j and k, we have

Eliminating α we have:

This gives us the Cartesian equation of a line.

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