3. (Unit Normal and Binormal Vector to a Curve) Let r(t) be a smooth vector function. This problem introduces two vectors that are orthogonal to the r'(t) unit tangent vector T(t) = |r' (t)| At any point where the curve defined by r(t) has nonzero curvature, the unit normal vector is defined by N(t) = T'(t) |T'(t)| a. Let v(t) be any vector function with |v(t)| =c, where c is a constant, for all t. Use the fact that |v(t)|2 = v(t) v(t) and the chain rule for dot product d (u(t) · v(t)) = u'(t) · v(t) + u(t) · v′(t) to show that v(t) and v'(t) are orthogonal. Hint: If you get stuck, see Example 4 of Section 13.2 in the textbook. b. Conclude that N(t) is orthogonal to T(t). c. Let r(t)=(sin 2t, - cos 2t, 4t). Compute T and N for this vector function r(t) at the point (0, 1, 2). d. Compute the vector B(t) T(t) x N(t). for the same vector function at the point (0, 1, 27). This is another unit vector which is orthogonal to T(t) called the binormal vector to the curve r(t). e. The plane generated by the normal and binormal vectors N and B at a point P on a curve C is called the normal plane of C at P. Find an equation for the normal plane of the same vector function at the point (0, 1, 2). Hint: This plane has normal vector the unit tangent vector T at the point (0, 1, 2). f. The plane generated by the unit tangent vector T and the normal vector N is called the oscu- lating plane of C at the point P. Find and equation for the osculating plane for the same vector function at the point (0, 1, 2). Hint: This plane has normal vector the binormal vector B at the point (0, 1, 2π).