Find the equation of the line passing through a point $B$, with position vector $b$ relative to an origin $O$, which is perpendicular to and intersects the line $r = a+\lambda c, c \neq 0$, given that $B$ is not a point of the line.
Finding the equation of an unknown vector that is perpendicular and intersects a line
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vectors
1 Answers
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Let $\textbf r_1$ be the point on the line which the perpendicular line through B intersects. Hence, we have that the vector $\textbf b - \textbf r_1$ must be perpendicular to the line, which has direction $\textbf c$. Hence, $$(\textbf b - \textbf r_1).\textbf c= 0$$ $$(\textbf b - (\textbf a + \lambda \textbf c)).\textbf c = 0$$ Assuming that $||\textbf c|| = 1$, we get $$\lambda = (\textbf b - \textbf a).\textbf c$$ Now , we have two points, $\textbf r_1$ and $\textbf b$. Line equation can be found