$\"\"$

(a)

Show that $\"\"$ is perpendicular to $\"\"$ .

Any vector function of constant magnitude is perpendicular to its derivative function for all $\"\"$ .

Thus, in perpendicular we know that $\"\"$ is perpendicular to $\"\"$ .But

$\"\"$

$\"\"$ ,

and so $\"\"$ is a scalar multiple of $\"\"$ .Thus, $\"\"$ must be perpendicular to $\"\"$ .

$\"\"$

(b)

Show that $\"\"$ is perpendicular to $\"\"$ .

$\"\"$ .

$\"\"$

$\"\"$ .

Therefore, $\"\"$ is perpendicular to $\"\"$ .

$\"\"$

(c)

Since $\"\"$ and $\"\"$ are not parallel at any time $\"\"$ , they deermine a plane at each value of $\"\"$ .Both $\"\"$ and $\"\"$ are orthogonal to that plane, so they must be parallel.Parallel vectors are scalar multiples of each other, so far each $\"\"$ there exists some value $\"\"$ that satisfies the equation.

$\"\"$

(d)

Since $\"\"$ , and since $\"\"$ and $\"\"$ always define the plane containing the curve, $\"\"$ must always have the same direction. Its also always a unit vector, so its magnitude and direction are fixed.This means that the vector is constant for all $\"\"$ , so that $\"\"$ .But then $\"\"$ .

Since $\"\"$ so it must be $\"\"$ for all $\"\"$ .

$\"\"$

(a) $\"\"$ is perpendicular to $\"\"$ .

(b) $\"\"$ is perpendicular to $\"\"$ .

(c) Since $\"\"$ and $\"\"$ are not parallel at any time $\"\"$ , they deermine a plane at each value of $\"\"$ .Both $\"\"$ and $\"\"$ are orthogonal to that plane, so they must be parallel.Parallel vectors are scalar multiples of each other, so far each $\"\"$ there exists some value $\"\"$ that satisfies the equation.

(d) $\"\"$ .