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Can someone please explain highlighted steps? $\color{red}{*}$

A direction vector of line is given by $$v=(1,-7,21)-(1,2,3) \Rightarrow (0,-9,18)$$

The equation of line is $$r=(1,2,3)+\lambda(0,-9,18)$$

Setting $t_0=1$ and $t_1=10$ and choosing $\color{red}{*}$

$$\lambda=\frac{t-t_0}{t_1-t_0} \Rightarrow \frac{t-1}{9}\color{red}{*}$$

gives the result: $$r=(1,2,3)+\frac{t-1}{9}(0,-9,18)$$

2 Answers 2

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I think $t_0$ represent initial point and $t_1$ point through which line is passing.

After that is formula to find $\lambda$.

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    Is that a formula to find $\lambda$? – 2017-02-06
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    Yes it is formula. – 2017-02-06
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    Ok thanks, I didn't see that formula before. – 2017-02-06
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    Maybe you are using other formula that involves points. – 2017-02-06
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    Maybe yes, In other problems I could calculate $\lambda$ with simple algebra. – 2017-02-06
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When $\lambda=0$, $r=(1,2,3)$, and when $\lambda=1$, $r=(1,-7,21)$. We replace $\lambda$ by $at+b$, so we have to find $a$ and $b$ such that when $t=t_0=1$, $\lambda=0$ and when $t=t_1=10$, $\lambda=1$. This gives you a pair of linear equations in the unknowns $a$ and $b$, from which simple algebra will lead you to the solution $a=\frac1{t_1-t_0}$, $b=-{t_0\over t_1-t_0}$, which means that $\lambda={t-t_0\over t_1-t_0}$.