00:01
In this question we are asked to find equations of the tangent plane and normal line to the surface z minus 6 is equal to x into e to the power y into cost z at the point negative 6 .0.
00:14
So let's start to solve this problem.
00:16
To find the equation of line and tangent plane first we find the normal vector to this surface.
00:24
The equation of surface is x into e to the power y minus z plus 6.
00:30
Equals to 0.
00:33
Let's say that this expression is equal to f.
00:35
X into e to the power by minus z plus 6 equals to 0.
00:40
Now normal vector to the surface is given by normal vector and vector is equal to gradient of f.
00:49
And for three dimension scalar field, gradient of that scalar field is given by i, dlf over dlx plus z, delf over dlx plus z, delf over dly, plus k, del f over del z equals to i, partial differentiation of this with respect to x is equal to e to the power y.
01:16
Plus z, partial differentiation of this with respect to y is equal to x into e to the power y plus k, partial differentiation of this with respect to z equals to negative 1.
01:30
When we take partial differentiation with respect to x, then variables other than x behave like a constant.
01:38
Similarly, for this partial differentiation, x and z views like k constant and for this partial differentiation x and y give like a constant.
01:54
N vector is equal to e to the power y i plus x into e to the power y z minus k.
02:02
Now normal vector at the point negative 6 .0 .0 is given by an vector at this point is equal to.
02:11
Now to find the normal vector at this point we put x equals to negative 6, y equals to 0 and z equal to 0.
02:18
E to the power 0 i plus negative 6 into e to the power 0 z minus k.
02:25
A to the power 0 is equal to 1 i minus 6 j minus k.
02:35
If a plane is tangent to this surface, then in that condition, the n vector is normal to the tangent plane.
02:50
Now we know if an vector is equal to a i plus b j plus ck is normal to a plane, then equation of that plane is given by ax plus by plus cz plus d equals to 0...