The movement of air on the surface of a certain closed controlled space is studied, and every point on the surface is associated with a vector reflecting the wind speed and direction at that point. The data obtained was seen to be effectively modeled by the conservative vector field F = (1+ ln (xy), %3D

Q: 1. Let ř(1) = . Find a vector and parametric equation of the line tangent to F(1) at the point (-…

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Q: Exercise 4. Show that if a has unit speed, i.e., |la'(t)|| = 1, then its velocity and acceleration…

A: We will prove the following result.

Q: QUESTION 10 Let r(t)=(1-t2, 5t, 2t3). Find the z-component of the vector r'(4).

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Q: A sodium ion (Na+) moves in the xy-plane with a speed of 2.90 ✕ 103 m/s. If a constant magnetic…

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Q: Sketch the curve with vector equation r(t) = (t2 – 2, t) in the xy-plane. Indicate with arrows the…

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Q: 7. Let b = (b1,..., bn) be a given vector and define-the function f(x) = f(x1, ..., Xn) = b. Show…

A: We have to solve given problem:

Q: Suppose that C has length 5. What is the value of F. dr if: (a) F(P) is normal to C at all points P…

A: Given: Length of C is 5. (a) F(P) is normal to C at all points P on C. ∫CF→·dr→=∫CFdrcos90°=0…

Q: Consider the function f(x,y) = 4x² - 5y² - 5 and the point (1,1). a. Find the unit vectors that give…

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Q: Determine a normal vector to the graph of the given function z = -2y³ - 3x² at the point (2, 1,…

A: Given: z = -2y3 - 3x2  Point: ( 2 , 1 , 14 )

Q: Question 1 Define the two vector valued functions of one variable u(t)= (-1/t, 2t, t²), v(t) := (t²,…

A: Let's find derivative of vector function.

Q: (9) Find a vector parameterizat1on of the line r(t) which passes through the points P = (2, –6, 4)…

A: The given points are P=(2,-6,4) and Q=(3,5,-3). We have to do vector parametrization. We subtract…

Q: If the position vector r(t)=e9t i +9sqrt(2)t j + e-9t k, compute the normal component of the…

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Q: Find the unit tangent vector (F(t)) and the principle unit normal vector (N(t) ) to the curve given…

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Q: (a) Sketch the plane curve with the given vector equation. (b) Find r'(1). (c) Sketch the position…

A: A.

Q: (a) Let vt) be a differentiable vector valued function of t. If v. (dy/dt) = 0 for all t, can we say…

A: The question is asking whether we can infer anything about the magnitude of a vector v(t), given…

Q: If an object travels in the xy-plane along the curve traced out by 
the vector function r(t) = (t…

A: We have to find the total distance traveled by the object from t = 0 to t = 4.

Q: If r(t) is the position vector of a particle in the plane at time t, find the acceleration vector.…

A: given,  r(t)=(6cost)i+(3sint)j