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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