(b) Let r(t)-ti- +(4-t) k, then the vector equation of the line tangent to the graph of r(e) at the point (4, 1,0) on the curve is A.r-(4i +j) + t(-4i+j+4k) B.r-(4i-3)+ t(-41 +j+ 4k) Cr-(4i+j) + t(4i +j+ 4k) D.r (4i+5) + t(-4i +j-4k)
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- Find a vector equation for the curve of intersection between the surfaces y^2-x^2=9 and 2x-3y-z=12.What is the vector parametric equation L(t) for the line through the points (5,-5,0) and (0,-3,3)?Suppose that in a certain region of space the electric potential V is given by V (x, y, z) = 8x ^ 2-7xy + 7xyz Find the rate of change of the potential in (-1,1, -1) in the direction of the vector v = 7i + 10j-8k.
- Find a vector equation for the tangent line to the curve of intersection of the cylinders x2+y2=25 and y2+z2=20 at the point (3, 4, 2).Find the directional derivative of P(x, y, z) = x ^ 3 - x * y ^ 2 - z at B(1,1,0) in the direction of vector A = 2i - 3j + 6kThe vector v = <a, 1, -1>, is tangent to the surface x2 + 2y3 - 3z2 = 3 at the point (2, 1, 1). Find a.
- Find the vector equation of the plane tangent to the surface z = x^3 - y^3 at the point (0,1,-1)The position of a particle is determined by the vector-valued function r(t)=<1-t^2, 3t, t^3>. Find the decomposition of the acceleration vector in terms of its tangential and normal components when the particle is at the point (0,3, 1).Find the equation of a line in vector form that is parallel to 3x−2y+z= 4, passes through the point (2,0,1), and is perpendicular to r(t) =〈2−t,2t,1 +t〉.
- Find the scalar equation of the plane that passes through point P(−3, 3, 5) and is perpendicular to the line of intersection of planes x + y − z − 2 = 0 and 2x − y + 3z − 1 = 0.write the equation of the plane with normal vector n passing through the given point in the scalar form ax + by + cz = d. n = (2, −4, 1), 1/3 , 2/3 , 1f(x,y) = 2x3+4y3-3x2-120x-192y+5 Is f(x,y) a curve, vector field, or surface?