Compute the outward flux of F(x, y, z) =
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Chapter 16 Solutions
Multivariable Calculus
- 219. The diagram at right shows a rectangular solid, two of whose vertices are A = (0, 0, 0) and G = (4,6,3). → (a) Find vector projections of AG onto AB, AD, and AE. (b) Find the point on segment AC that is closest to the midpoint of segment GH. 220. The result of reflecting across the line y = -x and then rotating 330 degrees counterclockwise around the origin is an F B Z E A G C H D -Yarrow_forwardFind a vector parametric equation for the ellipse that lies on the plane 4x – 4y + z = −7 and is inside the cylinder x² + y² = 9. ř(u, v) = = for 0 ≤ u ≤ 3 and 0 ≤ v ≤ 2π.arrow_forwardThe vector v = <a, 1, -1>, is tangent to the surface x2 + 2y3 - 3z2 = 3 at the point (2, 1, 1). Find a.arrow_forward
- 2. Find a vector that is perpendicular to the surface x² + y² + z² = xyz + 8 at the point (1,2,3).arrow_forwardFind the scalar equation of the plane that contain the line * = [2, 1,7] + t[0, 1, 0], t e R and is parallel to the line 7 = [3,0, 4] + s[2,–1,0], s e R. 1.arrow_forwardFind the orthogonal trajectories of the conicoid (x + y)z = 1 of the conics in which it is cut by the system of the plane x − y + z = K where K is a parametersarrow_forward
- Find a vector parallel to the line of intersection of the planes 5x−y+2z=4 and 4x−y−3z=0v→= (b) Show that the point (−1,−1,−1) lies on both planes. Then find a vector parametric equation for the line of intersection.r→(t)=arrow_forwardCompute foxy dx + (x³/2 + y³/2) dy where C is the square with vertices (0,0), (0,1), (1,1) and (1,0) oriented clockwisearrow_forward4. Consider the vector function r(z, y) (r, y, r2 +2y"). (a) Re-write this vector function as surface function in the form f(1,y). (b) Describe and draw the shape of the surface function using contour lines and algebraic analysis as needed. Explain the contour shapes in all three orthogonal directions and explain and label all intercepts as needed. (c) Consider the contour of the surface function on the plane z= for this contour in vector form. 0. Write the general equationarrow_forward
- Brazilian soccer star Marta has a penalty kick in the quarter-final match. She kicks the soccer ball from ground level with the (x, y)-coordinates (76, 21) on the soccer field shown in the figure and with initial velocity vo = 8i - 4j+23k ft/s. Assume an acceleration of 32 ft/s² due to gravity and that the goal net has a height of 8 ft and a total width of 24 ft. 105 ft 105 ft ri = rj = 165 ft rk = e 165 ft Determine the position function that gives the position of the ball t seconds after it is hit. (Use symbolic notation and fractions where needed.) r(t) = r(t)i + r(t)j +rk(t)k 12 12 Xarrow_forwardQ2: (a) Find the scalar and vector projection of i along ủ, where u ¯=(2, 3, 4) and v¯=(2, 2, 0). (b) Prove that the given planes are parallel or not, if yes then find the distance between them X + 3y + 2Z = 5, 2X +6y + 4Z = 11.arrow_forwardBrazilian soccer star Marta has a penalty kick in the quarter-final match. She kicks the soccer ball from ground level with the (x, y)-coordinates (76, 21) on the soccer field shown in the figure and with initial velocity vo = 8i - 4j+23k ft/s. Assume an acceleration of 32 ft/s² due to gravity and that the goal net has a height of 8 ft and a total width of 24 ft. 105 ft 1 105 ft r₁ = rj = rk = 165 ft Determine the position function that gives the position of the ball t seconds after it is hit. (Use symbolic notation and fractions where needed.) r(t) = ri(t)i + rj(t)j + rk(t)k Incorrect 81+76 -41+21 ↓ 23r+1672² 165 ft 12 12arrow_forward
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