The normal line to the surface x² + 2y² + ² = 25 at the point (1, -2, 4) is parallel to the vector O (1,4,-4) O (1,-4,4) O (2,8,-4) O (2,-8,4)
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- Find a vector tangent to the curve of intersection of the two cyclinders x2+y2=32x2+y2=32 and y2+z2=32y2+z2=32 at the point (−4,−4,4)(−4,−4,4).Find a unit vector n that is normal to the surface 2z^2−x^4−y^4=1 at P=(1,2,3) that points in the direction of the xy-plane (in other words, if you travel in the direction of n, you will eventually cross the xy-plane).Find an equation of the plane through (1, −3, 5) with normal vector n =〈2, 1, −4〉.
- Find a unit vector tangent to the curve of intersection of -x^2 - 2y^3 = z - 3 and 25/x2 - 4y - 3z^2 = - x + 6 at the point (1,1,0).Find an equation of the plane passing through the point (1,4,2) with a normal vector of the form n=a,b,1 that slices off the region in the first octant with the least volume.Find a set of symmetric equations for the normal line to the surface xyz = 12 at the point (2, −2, −3).
- find a vector parametrization for the line with thegiven description. Passes through (1, 1, 1) parallel to the line through (2, 0, −1) and(4, 1, 3)Determine the general equation of the plane that passes through the point P (-1,1,0) and that has a normal vector n = (2,0, -4).Realize the Cauchy-Schwarz inequality for the vectors x= ( -1, 2, 4,3), y=(0,-2,1, 4) ∈ R4 and find the angle between these vectors.