Question
Two parameterized lines are given. Are they the same line?$$\begin{aligned}&\vec{r}_{1}(t)=(5-3 t) \vec{i}+2 t \vec{j}+(7+t) \vec{k}\\&\vec{r}_{2}(t)=(5-6 t) \vec{i}+4 t \vec{j}+(7+3 t) \vec{k}\end{aligned}$$
Step 1
For line 1, the direction vector is given by the coefficients of t, which is <-3, 2, 1>. For line 2, the direction vector is given by the coefficients of t, which is <-6, 4, 3>. Now, let's check if these direction vectors are proportional. If they are, then the Show more…
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Two parameterized lines are given. Are they the same line? $$\begin{aligned} &\vec{r}_{1}(t)=(5-3 t) \vec{i}+(1+t) \vec{j}+2 t \vec{k}\\ &\vec{r}_{2}(t)=(2+6 t) \vec{i}+(2-2 t) \vec{j}+(2-4 t) \vec{k} \end{aligned}$$
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