5. Find a parametrization (line segment) for the line segment between points $$ A $$ and $$ B $$. a) $$ A(1,1,1), B(2,2,4) $$ b) $$ A(2,1,2), B(2,2,-3) $$ c) $$ A(2,1,-2), B(1,4,1) $$

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UpStudy AI · Accepted Answer

**(a) For $$ A(1,1,1) $$ and $$ B(2,2,4) $$:**

Let $$ t $$ be a parameter where $$ t \in [0,1] $$. The line segment can be described by the equation
$$
\mathbf{r}(t) = (1-t) \, \mathbf{A} + t \, \mathbf{B}.
$$
Substituting the coordinates of $$ A $$ and $$ B $$:
$$
\begin{aligned}
\mathbf{r}(t) &= (1-t)(1,\,1,\,1) + t(2,\,2,\,4) \
&= (1-t + 2t,\, 1-t + 2t,\, 1-t + 4t) \
&= (1+t,\, 1+t,\, 1+3t).
\end{aligned}
$$
Thus, the parametrization is:
$$
x(t)=1+t,\quad y(t)=1+t,\quad z(t)=1+3t,\quad \text{where } t \in [0,1].
$$

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**(b) For $$ A(2,1,2) $$ and $$ B(2,2,-3) $$:**

Using the same parametrization method:
$$
\mathbf{r}(t) = (1-t)(2,\,1,\,2) + t(2,\,2,-3).
$$
Calculating each coordinate:
$$
\begin{aligned}
x(t) &= (1-t)(2) + t(2) = 2, \
y(t) &= (1-t)(1) + t(2) = 1-t+2t = 1+t, \
z(t) &= (1-t)(2) + t(-3) = 2-2t-3t = 2-5t.
\end{aligned}
$$
The parametrization is:
$$
x(t)=2,\quad y(t)=1+t,\quad z(t)=2-5t,\quad \text{where } t \in [0,1].
$$

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**(c) For $$ A(2,1,-2) $$ and $$ B(1,4,1) $$:**

Again, using
$$
\mathbf{r}(t) = (1-t)(2,\,1,-2) + t(1,\,4,\,1),
$$
we compute the coordinates:
$$
\begin{aligned}
x(t) &= (1-t)\cdot 2 + t\cdot 1 = 2-2t+t = 2-t, \
y(t) &= (1-t)\cdot 1 + t\cdot 4 = 1-t+4t = 1+3t, \
z(t) &= (1-t)(-2) + t\cdot 1 = -2+2t+t = -2+3t.
\end{aligned}
$$
Thus, the parametrization is:
$$
x(t)=2-t,\quad y(t)=1+3t,\quad z(t)=-2+3t,\quad \text{where } t \in [0,1].
$$
**(a) For points $$ A(1,1,1) $$ and $$ B(2,2,4) $$:** $$ x(t) = 1 + t,\quad y(t) = 1 + t,\quad z(t) = 1 + 3t,\quad \text{where } t \in [0,1]. $$ **(b) For points $$ A(2,1,2) $$ and $$ B(2,2,-3) $$:** $$ x(t) = 2,\quad y(t) = 1 + t,\quad z(t) = 2 - 5t,\quad \text{where } t \in [0,1]. $$ **(c) For points $$ A(2,1,-2) $$ and $$ B(1,4,1) $$:** $$ x(t) = 2 - t,\quad y(t) = 1 + 3t,\quad z(t) = -2 + 3t,\quad \text{where } t \in [0,1]. $$

Find a parametrization (line segment) for the line segment between points and .
a)
b)
c)

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Find a parametrization (line segment) for the line segment between points and . a) b) c)