Represent the line segment from P to Q by a vector-valued function and by a set of parametric equations P(−3, −6, −1), Q(−1, −9, −8).

emancipezN

emancipezN

Answered question

2021-02-25

Represent the line segment from P to Q by a vector-valued function and by a set of parametric equations P(3,6,1),Q(1,9,8).

Answer & Explanation

un4t5o4v

un4t5o4v

Skilled2021-02-26Added 105 answers

Two vectors P(x1,y1,z1)andQ(x2,y2,z2) Line segment P to Q : 1) Parametric form is x=x1+at
y=y1+bt
z=z1+ct
Where a=x2x1,b=y2y1,c=z2z1 2) vector valued form is r=<x,y,z> Given vectors P(3,6,1)andQ(1,9,8) Line segment P to Q: Firstly evaluate a, b and c PQ=<x2x1,y2y1,z2z1>
=<(1)(3),(9)(6),(8)(1)>
=<1+3,9+6,8+1>
=<2,3,7> 1) Parametric form is x=x1+at
=3+(2)t
=3+2t
y=y1+bt
=6+(3)t
=63t
z=z1+ct
=1+(7)t
=17t 2) Vector valued function: r(t)=<3+2t,63t,17t>
(3+2t)i+(63t)j+(17t)k Hence 1) vector valued function is r(t)=(3+2t)i+(63t)j+(17t)k 2) Parametric form is x=3+2t
y=63t
z=17t
nick1337

nick1337

Expert2023-05-29Added 777 answers

To represent the line segment from P to Q using a vector-valued function and a set of parametric equations, we can use the following formulations:
1. Vector-Valued Function:
𝐫(t)=𝐏+t(𝐐𝐏)
2. Parametric Equations:
x=xP+t(xQxP),
y=yP+t(yQyP),
z=zP+t(zQzP),
where 𝐏(3,6,1) and 𝐐(1,9,8) are the given points, and t is a parameter representing the position along the line segment.
Vasquez

Vasquez

Expert2023-05-29Added 669 answers

Answer:
x(t)=3+2ty(t)=63tz(t)=17t
Explanation:
Let's start with the vector-valued function. We can define a position vector 𝐫(t) that represents any point on the line segment. The vector 𝐫(t) will have components x(t), y(t), and z(t), which are functions of the parameter t.
Using the two given points, we can find the direction vector of the line segment by subtracting the coordinates of P from the coordinates of Q. The direction vector will give us the change in position as we move along the line segment.
Let's denote the direction vector as 𝐯:
𝐯=(1(3)9(6)8(1))=(237)
Now, we can write the vector-valued function as:
𝐫(t)=(361)+t(237)
Next, let's find the set of parametric equations for the line segment. We can express the coordinates x, y, and z in terms of the parameter t:
x(t)=3+2t
y(t)=63t
z(t)=17t
Thus, the set of parametric equations for the line segment is:
x(t)=3+2ty(t)=63tz(t)=17t
RizerMix

RizerMix

Expert2023-05-29Added 656 answers

Let's denote the vector-valued function as 𝐫(t) and the parametric equations as x(t), y(t), and z(t).
The direction vector, 𝐝, is given by:
𝐝=𝐐𝐏
Substituting the given coordinates, we have:
𝐝=(198)(361)=(1(3)9(6)8(1))=(237)
Now, we can represent the line segment using the vector-valued function:
𝐫(t)=𝐏+t𝐝
Substituting the coordinates and the direction vector, we get:
𝐫(t)=(361)+t(237)=(3+2t63t17t)
The line segment can also be represented by the set of parametric equations:
x(t)=3+2ty(t)=63tz(t)=17t
These equations describe the coordinates of points on the line segment as a function of the parameter t. By varying the value of t within a suitable range, we can traverse the line segment from point P to point Q.

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?