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- find a parametrization for the curve. the ray (half line) with initial point (-1, 2) that passes through the point (0, 0)At a certain moment, a moving particle has velocity v = (2, 2, −1)and a = (0, 4, 3). Find T, N, and the decomposition of a into tangentialand normal components.Find parametric equations for the normal line to the surfacez=ln(3x2 +7y2 +1) at the point (0,0,0)
- 1. The distance of a point in the 3-D system from the origin a. is defined by the absolute value of the vector from the origin to this point. b. is the square root of the square of the sums of the x-, y- and z-values. c. is the square root of the sum of the squares of x-, y- and z-values. d. can either be negative or positive. e. None of the above. 2. In parametrizing lines connected by two points in 3-D plane, a. there is only one correct parametrization. b. symmetry equations may not exist. c. a, b, and c must not be equal to 0. d. the vector that connects the two points is a scalar multiple of the vector containing the direction numbers. e. None of the above. 3. A plane in 3D-space system a. is generated by at least three points. b. can lie in more than one octant. c. must have a z-dimension. d. must have a point other than the origin. e. None of the above. 4. A quadric surface a. must have either x2, y2, or z2 or a combination of those, on its general expression. b. must have a…Obtain the differential equation of the family of plane curves described. 1. All ellipses having its centers at the origin and traverse axis x.Double integrate under z=xy, above the triangle with vertices (0,1),(0,4),(1,1).
- or this problem, consider a particle traveling within the force field F = < -y,x,1/2 > along the parametrized curve r(t) = < t cos(t),t sin(t),1/2t > from the point (0,0,0) to the point (2pi,0,pi) Explain why the work done moving the particle along the path in this force field is positive. Compute the work done on a particle traveling along the given parametrized curve within the force field.The flow lines (or streamlines) of a vector field are the paths followed by a particle whose velocity field is the given vector field. Thus the vectors in a vector field are tangent to the flow lines. (a) Use a sketch of the vector field F(x, y) = xi − yj to draw some flow lines. From your sketches, can you guess the equations of the flow lines? (b) If parametric equations of a flow line are x = x(t), y = y(t), explain why these functions satisfy the differential equations dx/dt = x and dy/dt = −y. (c) Solve the differential equations to find an equation of the flow line that passes through the point (x, y) = (−1, −1).Find a parametrization for the path that travels along the graph of y=sinx from (0,0) to (pi,0).