Chapter 14: Q 18. (page 1106)

Give a smooth parametrization $r (t)$ for the straight line between the points $(π, e, 1)$ and $(3, 7, 13)$ .

Short Answer

Expert verified

The required parametrization is $r (t) = ⟨ π + (3 - π) t, e + (7 - e) t, 1 + 12 t ⟩, 0 \leq t \leq 1$

Step by step solution

Given Information

Smooth Parametrization $r (t)$ for straight line between the points $(π, e, 1)$ and $(3, 7, 13)$ .

Write parametric form of the equation of a line segment with two points

The parametric form of the equation of a line segment with two points $(x_{1}, y_{1}, z_{1})$ and $(x_{2}, y_{2}, z_{2})$ is

$x = (1 - t) x_{1} + t x_{2}, y = (1 - t) y_{1} + t y_{2}, z = (1 - t) z_{1} + t z_{2}$

Now,

$(x_{1}, y_{1}, z_{1}) = (π, e, 1)$ and $(x_{2}, y_{2}, z_{2}) = (3, 7, 13)$

$\Rightarrow x = (1 - t) x_{1} + t x_{2}$

$= (1 - t) (π) + t (3)$

$= π - π t + 3 t$

$= π + (3 - π) t$

Complete the Parametrization

Also,

$y = (1 - t) y_{1} + t y_{2}$

$= (1 - t) (e) + t (7)$

$= e - e t + 7 t$

$= e + (7 - e) t$

And

$z = (1 - t) z_{1} + t z_{2}$

$= (1 - t) (1) + t (13)$

$= 1 - t + 13 t$

$= 1 + 12 t$

The parametric form is

$x = π + (3 - π) t, y = e + (7 - e) t, z = 1 + 12 t$

Hence,

$r (t) = ⟨ π + (3 - π) t, e + (7 - e) t, 1 + 12 t ⟩, 0 \leq t \leq 1$

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Give a smooth parametrization $r (t)$ for the straight line between the points $(π, e, 1)$ and $(3, 7, 13)$ .

Short Answer

Step by step solution

Given Information

Write parametric form of the equation of a line segment with two points

Complete the Parametrization

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Give a smooth parametrizationr(t) for the straight line between the pointsπ,e,1and(3,7,13).

Short Answer

Step by step solution

Given Information

Write parametric form of the equation of a line segment with two points

Complete the Parametrization

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Mechanics Maths

Logic and Functions

Discrete Mathematics

Geometry

Pure Maths

Study anywhere. Anytime. Across all devices.

Give a smooth parametrization $r (t)$ for the straight line between the points $(π, e, 1)$ and $(3, 7, 13)$ .