Let L be the line with parametric equations x=-5 y=-6- z=9-t Find the vector equation for a line that passes through the point P=(-3, 10, 10) and intersects L at a point that is distance 5 from the point Q=(-5, -6, 9). Note that there are two possible correct answers. Use the square root symbol 'V' where needed to give an exact value for your answer. 8 N
Consider the following function 6 f(x, y,z)=z - x? cos(my) + xy? (i) Find the gradient of the function f(x, y, z) at the point P,(2,-1,-7). (ii) Find the directional derivative of f(x, y, z) at P,(2,-1,-7) along the direction of the vector ū = 2î+j+2k. (iii) Find the equation of the tangent plane to the surface given below at the point P,(2,-1, -7). 6 :- xcos(ty) + = 0 xy
18. Consider the line L with vector equation (x, y, z)-(3, 4,-1 1,-2, 5) and the point P(2, 5, 7). Show that P is not on L, and then find a Cartesian equation for the plane that contains both P and L.
The coordinates of a bird flying in the xy-plane are given by x(t)=αt and y(t)=3.0m−βt2, where α=2.4m/s and β=1.2m/s2 1-Calculate the velocity vector of the bird as a function of time. Give your answer as a pair of components separated by a comma. For example, if you think the x component is 3t and the y component is 4t, then you should enter 3t,4t. Express your answer using two significant figures for all coefficients 2-Calculate the acceleration vector of the...
The temperature at a point (x, y, z) is given by T(x, y, z) = 10e e-3x2 – 3y2 – 2z2 In which direction does the temperature increase fastest at the point (3, 1, 4)? Express your answer as a UNIT vector.
The temperature at a point (x, y, z) is given by T(x, y, z) = 10e e-3x2 – 3y2 – 2z2 In which direction does the temperature increase fastest at the point (3, 1, 4)? Express your answer as a UNIT vector.
(1 point) Consider the line L(t) = (2+ 3t, 6-t). Then L intersects: 1. The X-axis at the point (2,6) when t = 0 2. The y-axis at the point (2,6) when t = 0 3. The parabola y = x2 at the points and when t = and
The temperature at a point (x, y, z) is given by T(x, y, z) = 100e-x2 - 5y2 - 722 where Tis measured in °C and x, y, z in meters. (a) Find the rate of change of temperature at the point P12,-1, 3) in the direction towards the point (5, -3, 6). °C/m (b) In which direction does the temperature increase fastest at P? (c) Find the maximum rate of increase at P.
1) Show that two lines are skew x+1 y+2 z+3 4:x=y=z and L: +7=5 2) Find the general equation of the plane containing the point P (1,2,3 ) and L, . 3) Find the point Q-the point of intersection the plane found in 2) and the line L. 4) Find the distance from the point (1,-1,2) to the line Lą.
(8 points) The temperature at a point (x, y, z) is given by T(x, y, z) = 1300e-x-2y-2? where T is measured in °C and x, y, and z in meters. 1. Find the rate of change of the temperature at the point P(2, -1, 2) in the direction toward the point Q(3,-3,3). Answer: Dp S(2.-1, 2) = 2. In what direction does the temperature increase fastest at P? Answer: 3. Find the maximum rate of increase at P. Answer: