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- Find both the parametric and vector equation of the line segment between (−2, 3) and (5, −1) where 0 ≤ t ≤ 1. Please show all your work.Find the cartesian equation of the straight line tangent to the plane curve given by the vector equation r(t) = (t- 6) i + + 3t - 2) į. at the point P(-5,4) on the curve. Express your answer in the form y = ax + b , where a and b are integers. equarion of the tangent line is given by y = 9x + 49For the curve y=sqrt x-5 (a) Find a vector parallel to the tangent line to the curve at the point (9, 2). (b) Find a vector orthogonal to the tangent line to the curve at the point(9,2).
- Find the new equation of the line y=x after applying the translation by the vector (2, -1)Use vectors to show that the distance from P1(x1, y1) to the line ax + by = c is d = |ax1 + by1 - c|/(a2 + b2)^1/2 .Let / denote the line in R² with equation y = 3x - 2. Let Q = (2, -1). (a) Write the line / in vector form (i.e. of the form p = po + td where Po is a point on the line and d is a direction vector). (b) Find the distance from Q to 1. (c) Find the point on I closest to Q.
- Create the vector & parametric equations for the line that passes through the point L₁ x 3- 2t (0, 1, -2) and is parallel to the line y z = = - 5t - 1-t.Determine the equation of the tangent plane and a vector equation of the normal line to x²y - 4ze+y +35= 0 at (3,-3,2). A. tangent plane: -26x + y −4z +89 = 0, normal line: (3+26t, -3-t, 2+4t) B. tangent plane: -26x + y - 4z +89 = 0, normal line: (3-26t, -3+t, 2-4t) C. tangent plane: 26x + y − 4z +89 = 0, normal line: (3+26t, -3-t, 2+4t) D. tangent plane: 26x + y - 4z +89 = 0, normal line: (3-26t, -3+t, 2-4t)Find T, N, and B for the curve r(1) = (171, 171,e') at (0,0, 1). Hint: After finding T', plug in the specific value for 1 before computing N and B. (Use symbolic notation and fractions where needed. Enter vectors in the form (*, *, *).) T = N = B =
- Parameterize the line from (1,5) to (-4, -3) so that the line isat (1, 5) at t=0 and at (-4,-3) at t=1. y= Your parametric equation should be linear functions of t.Let the velocity vector be v(t)=⟨5t^4,−3sin(t),4exp(2t)⟩ and the initial position vector be r(0)=⟨−1,6,−1⟩. Compute the position vector r(t).