Question

(1 point) Suppose F⃗ (x,y)=−yi⃗ +xj⃗ F→(x,y)=−yi→+xj→ and CC is the line segment from point P=(4,0)P=(4,0) to Q=(0,5)Q=(0,5). (a) Find a vector parametric equation r⃗ (t)r→(t) for the line segment CC so that points PP and QQ correspond to t=0t=0 and t=1t=1, respectively. r⃗ (t)=r→(t)= <4,0>+t<-4,5> (b) Using the parametrization in part (a), the line integral of F⃗ F→ along CC is ∫CF⃗ ⋅dr⃗ =∫baF⃗ (r⃗ (t))⋅r⃗ ′(t)dt=∫ba∫CF→⋅dr→=∫abF→(r→(t))⋅r→′(t)dt=∫ab 20 dtdt with limits of integration a=a= 0 and b=b= 1 (c) Evaluate the line integral in part (b). 20 (d) What is the line integral of F⃗ F→ around the clockwise-oriented triangle with corners at the origin, PP, and QQ

Answer 1

The vector parametric equation for the line segment CC is <4,0> + t<-4,5>. The line integral of F⃗ along CC is 20. The line integral of F⃗ around the clockwise-oriented triangle with corners at the origin, P, and Q cannot be determined without additional information.

Step-by-step explanation:

(a) To find a vector parametric equation for the line segment CC, we can use the points P=(4,0) and Q=(0,5). We can represent the line segment CC as r(t) = <4,0> + t<-4,5>, where t is the parameter. This equation represents the line segment from P to Q, with t=0 corresponding to P and t=1 corresponding to Q.

(b) Using the parametrization in part (a), we can evaluate the line integral of F⃗ along CC. The line integral is given by ∫CF⃗ ⋅ dr⃗ = ∫baF⃗ (r⃗ (t))⋅r⃗ ′(t)dt. In this case, the line integral is ∫01-5yi⃗ +4xj⃗⋅-4i⃗ +5j⃗ dt

(c) Evaluating the line integral from part (b), we get 20.

(d) The line integral of F⃗ around the clockwise-oriented triangle with corners at the origin, P, and Q can be found using the Green's theorem. We can calculate it by subtracting the line integral along CP from the line integral along CQ. However, we would need more information to determine the path from C to the origin.