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Find the equation of the line passing through a point $B$, with position vector $\mathbf{b}$ relative to the origin $O$, which is perpendicular to and intersects the line $\mathbf{r} = \mathbf{a} + \lambda \mathbf{c}$ where $\mathbf{c} \ne \mathbf{0}$, given that $B$ is not a point of the line.

No idea how to approach question. Is it that the dot product of line passing through $B$ and $\mathbf{r} =\mathbf{0}$?

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    My, my, you are a slow learner, dear Shem. **Don't abuse this site and/or its users.** – 2017-02-20

1 Answers 1

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If $B$ is not on $\mathbf{r}=\mathbf{a}+\lambda \, \mathbf{c}$, then $\mathbf{b} \ne \mathbf{a}$.

The point of intersection is given by

\begin{align*} 0 &= (\mathbf{r}-\mathbf{b}) \cdot \mathbf{c} \\ 0 &= \mathbf{a} \cdot \mathbf{c}+\lambda \, c^2-\mathbf{b} \cdot \mathbf{c} \\ \lambda &= \frac{(\mathbf{b}-\mathbf{a}) \cdot \mathbf{c}}{c^2} \end{align*}

So the equation of the required line is

$$\mathbf{r}= \mathbf{b}+\mu \left[ \mathbf{a}-\mathbf{b}+ \frac{(\mathbf{b}-\mathbf{a}) \cdot \mathbf{c}}{c^2} \, \mathbf{c} \right]$$

or equivalently in case of $\mathbb{R}^3$,

$$\mathbf{r}= \mathbf{b}+\frac{\mu \, \mathbf{c} \times [(\mathbf{a}-\mathbf{b}) \times \mathbf{c}]}{c^2}$$

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    For your last equation, how do you know that you‘re working in $\mathbb R^3$, so that cross products are defined? – 2017-02-15
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    @amd Just in case – 2017-02-15
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    Thanks alot, can i just ask why is the required line [a-b+......} just confused about the -b part – 2017-02-18
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    It passes through $B$ pointing in direction $\mathbf{a}+\lambda \, \mathbf{c}-\mathbf{b}$. – 2017-02-18