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).

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un4t5o4v · Accepted Answer

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=x2−x1,b=y2−y1,c=z2−z1 2) vector valued form is r=&amp;lt;x,y,z&amp;gt; Given vectors P(−3,−6,−1)andQ(−1,−9,−8) Line segment P to Q: Firstly evaluate a, b and c  ⎯⎯⎯PQ=&amp;lt;x2⎯x1,y2⎯y1,z2⎯z1&amp;gt;  ⎯⎯⎯⎯⎯⎯⎯⎯⎯=&amp;lt;(⎯1)⎯(⎯3),(⎯9)⎯(⎯6),(⎯8)⎯(⎯1)&amp;gt;  ⎯⎯⎯=&amp;lt;⎯1+3,⎯9+6,⎯8+1&amp;gt;  ⎯⎯=&amp;lt;2,⎯3,⎯7&amp;gt; 1) Parametric form is  x=x1+at  ⎯=⎯3+(2)t  ⎯=⎯3+2t   y=y1+bt  ⎯⎯=⎯6+(⎯3)t   ⎯⎯=⎯6⎯3t   z=z1+ct   ⎯⎯=⎯1+(⎯7)t   ⎯⎯=⎯1⎯7t 2) Vector valued function: ⎯⎯⎯⎯⎯r(t)=&amp;lt;⎯3+2t,⎯6⎯3t,⎯1⎯7t&amp;gt;  ⎯⎯⎯⎯⎯(⎯3+2t)i+(⎯6⎯3t)j+(⎯1⎯7t)k Hence 1) vector valued function is  ⎯⎯⎯⎯⎯r(t)=(⎯3+2t)i+(⎯6⎯3t)j+(⎯1⎯7t)k 2) Parametric form is  ⎯x=⎯3+2t  ⎯⎯y=⎯6⎯3t   ⎯⎯z=⎯1⎯7t

nick1337 · Accepted Answer

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(𝐐−𝐏)2. Parametric Equations:x=xP+t(xQ−xP),y=yP+t(yQ−yP),z=zP+t(zQ−zP),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 · Accepted Answer

Answer:x(t)=−3+2ty(t)=−6−3tz(t)=−1−7tExplanation:Let&#039;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&#039;s denote the direction vector as 𝐯:𝐯=(−1−(−3)−9−(−6)−8−(−1))=(2−3−7)Now, we can write the vector-valued function as:𝐫(t)=(−3−6−1)+t(2−3−7)Next, let&#039;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+2ty(t)=−6−3tz(t)=−1−7tThus, the set of parametric equations for the line segment is:x(t)=−3+2ty(t)=−6−3tz(t)=−1−7t

RizerMix · Accepted Answer

Let&#039;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=𝐐−𝐏Substituting the given coordinates, we have:𝐝=(−1−9−8)−(−3−6−1)=(−1−(−3)−9−(−6)−8−(−1))=(2−3−7)Now, we can represent the line segment using the vector-valued function:𝐫(t)=𝐏+t𝐝Substituting the coordinates and the direction vector, we get:𝐫(t)=(−3−6−1)+t(2−3−7)=(−3+2t−6−3t−1−7t)The line segment can also be represented by the set of parametric equations:x(t)=−3+2ty(t)=−6−3tz(t)=−1−7tThese 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.

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).

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