Find the equation to the tangent as the point u on the circular helix x=acos u y=asinu z=cuSee answer
Find the equation to the tangent as the point u on the circular helix x acos u y asinu z cu
Question
Basic Answer
Step 1: Find the derivatives
To find the equation of the tangent line, we first need to find the derivatives of the parametric equations of the helix with respect to u:
dx/du = -a sin u
dy/du = a cos u
dz/du = c
These derivatives give us the direction vector of the tangent line at any point u.
Step 2: Find the point on the helix
The point on the helix at parameter u is given by (a cos u, a sin u, cu).
Step 3: Construct the tangent line equation
The equation of a line in 3D space can be written in vector form as:
(x, y, z) = (x₀, y₀, z₀) + t(dx/du, dy/du, dz/du)
where (x₀, y₀, z₀) is a point on the line and (dx/du, dy/du, dz/du) is the direction vector. Substituting the values we found:
(x, y, z) = (a cos u, a sin u, cu) + t(-a sin u, a cos u, c)
Step 4: Separate into individual equations
We can separate the vector equation into three scalar equations:
x = a cos u – at sin u
y = a sin u + at cos u
z = cu + ct
Final Answer
The equation of the tangent line to the circular helix at the point u is given by:
x = a cos u – at sin u
y = a sin u + at cos u
z = cu + ct
where ‘t’ is a parameter.