Equation of a Line in Vector Form

Here you will learn equation of a line in vector form passing through a fixed point and passing through two points.

Let’s begin –

Equation of a Line in Vector Form

The vector equation of a straight line passing through a fixed point with position vector \(\vec{a}\) and parallel to a given vector \(\vec{b}\) is 

\(\vec{r}\) = \(\vec{a}\) + \(\lambda \vec{b}\), where \(\lambda\) is scalar.

Note : In the above equation \(\vec{r}\) is the position vector of any point P (x, y, z) on the line. Therefore, \(\vec{r}\) = \(x\hat{i} + y\hat{j} + z\hat{k}\).

Example : Find the vector equation of a line which passes through the point with position vector \(2\hat{i} – \hat{j} + 4\hat{k}\) and is in the direction \(\hat{i} + \hat{j} – 2\hat{k}\).

Solution : Here \(\vec{a}\) = \(2\hat{i} – \hat{j} + 4\hat{k}\) and \(\vec{b}\) = \(\hat{i} + \hat{j} – 2\hat{k}\).

So, the vector equation of the required line is

\(\vec{r}\) = \(\vec{a}\) + \(\lambda \vec{b}\)

or, \(\vec{r}\) = (\(2\hat{i} – \hat{j} + 4\hat{k}\)) + \(\lambda (\hat{i} + \hat{j} – 2\hat{k})\), where \(\lambda\) is a scalar.

Equation of Line in Vector Form Passing Through Two Points

The vector equation of line passing through two points with position vectors \(\vec{a}\) and \(\vec{b}\) is

\(\vec{r}\) = \(\lambda\) \((\vec{b} – \vec{a})\), where \(\lambda\) is a scalar

Example : Find the vector equation of a line which passes through the point A (3, 4, -7) and B (1, -1, 6)

Solution : We know that the vector equation of line passing through two points with position vectors \(\vec{a}\) and \(\vec{b}\) is,

\(\vec{r}\) = \(\lambda\) \((\vec{b} – \vec{a})\)

Here \(\vec{a}\) = \(3\hat{i} + 4\hat{j} – 7\hat{k}\) and \(\vec{b}\) = \(\hat{i} – \hat{j} + 6\hat{k}\).

So, the vector equation of the required line is

 \(\vec{r}\) = (\(3\hat{i} + 4\hat{j} – 7\hat{k}\)) + \(\lambda\)  (\(\hat{i} – \hat{j} + 6\hat{k}\) – \(3\hat{i} + 4\hat{j} – 7\hat{k}\))

or, \(\vec{r}\) = (\(3\hat{i} + 4\hat{j} – 7\hat{k}\)) + \(\lambda\) (\(-2\hat{i} – 5\hat{j} + 13\hat{k}\))

where \(\lambda\) is a scalar.

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