Expert Answer to "Sketch the curve represented by the vector-valued function..."

Antinazius

Answered question

2021-11-25

Sketch the curve represented by the vector-valued function

r (t) = (t + 1) i + (3 t - 1) j + 2 t k

and give the orientation of the curve.

Answer & Explanation

Kathleen Ashton

Beginner2021-11-26Added 15 answers

Step1
Given:
The vector-valued function $r (t) = (t + 1) i + (3 t - 1) j + 2 t k$ and it gives the orientation of the curve.
Step2
As shown below, the vector-valued function is
$r (t) = (t + 1) i + (3 t - 1) j + 2 t k$
Below is a list of the first two parametric equations
$x = t + i and y = 3 t - 1$
The two parametric equations are determined by the rectangular equation
The value of t from x=t+ 1 is calculated as below,
$x = t + i$
$x - 1 = t + 1 - 1$
$t = x - 1$
The value of t=x-1.
Step3
On Substituting the value of t in y= 3t -1 to get
$y = 3 (x - 1) - 1$
$= 3 x - 3 - 1$
$= 3 x - 4$
Hence, the value of y=3x-4.
Thus, the rectangular equation is $y = 3 x - 4$
The curve is therefore a straight line that is determined in relation to the space
The third parametric equation, z= 2t, is produced as a result of the curve moving upward as the value of t increases
The vector valued function of the graph is being represented and the orientation is shown in the graph as,
$r (t) = (t + 1) i + (3 t - 1) j + 2 t k$

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