Represent the line segment from P(−2, −3, 8), Q(5, 1, −2) by a vector-valued function and by a set of parametric equations.

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Layton · Accepted Answer

We have, points P(−2, −3, 8) and Q(5, 1, −2) Now, the component vector from P to Q , v(t)=&amp;lt;5−(−2),1−(−3),−2−8&amp;gt;⇒v(t)=&amp;lt;7,4,−10&amp;gt; So, the direction number are a=7,b=4 and c=−10 For initial point on the line, use point P(−2,−3,8) We get,x0=−2,y0=−3andz0=8 Now, to form the parametric equation: x=x0+at,y=y0+btandz=z0+ct…….(1) Putting the value of a=7,b=4,c=−10 and x0=−2,y0=−3z0=8 in eq.(1),we get x=−2+7t,y=−3+4t and z=8−10t Now, to form a vector−valued function: r(t)=x(t)i+y(t)j+z(t)k We have, x=−2+7t,y=−3+4t and z=8−10t  So, r(t)=(7t−2)i+(4t−3)j+(8−10t)k Hence, set of parametric equations be ,x=−2+7t,y=−3+4t and z=8−10t and vector−valued function be, r(t)=(7t−2)i+(4t−3)j+(8−10t)k

Represent the line segment from P(−2, −3, 8), Q(5, 1, −2) by a vector-valued function and by a set of parametric equations.

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