91Ó°ÊÓ

How must \(\mathbf{a}, \mathbf{b}, \mathbf{c}\) be related in order to satisfy the associative law \((\mathbf{a} \times \mathbf{b}) \times \mathbf{c}\) \(=\boldsymbol{a} \times(\mathbf{b} \times \mathbf{c}) ?\)

When a particle moves in a circular orbit (like a stone swung at the end of a string), its position vector from the center has constant length. In which direction is its velocity? If its speed is constant, its velocity is a vector of constant length. In which direction is its acceleration?

The orthogonal projection of the circular helix on a plane through its axis, such as the plane \(x=0\), is the sine curve

Find the evolute of the cycloid $$ x=u+\sin u_{+} \quad y=1+\cos u . $$

Find \(\kappa\) and \(\tau\) for the curve $$ x=3 u-u^{3}, \quad y=3 u^{2}, \quad z=3 u+u^{3}, $$ and deduce that this curve is a helix.

Find the involute of the circle $$ x=\cos u_{1} \quad y=\sin u, $$ beginning at the point where \(u=0\).

Simplify \((\mathbf{a} \times \mathbf{b}) \times(\boldsymbol{c} \times \mathbf{d})\) two ways and, by equating the results, deduce an identity connecting four vectors such as [a b c] d.

Describe the surface formed by the midpoints of all the chords of a circular helix.

Simplify \((\mathbf{a} \times \mathbf{b}) \cdot(\boldsymbol{a} \times \mathbf{b})\), and show that the result could have been anticipated in virtue of a well known trigonometrical identity.

From "simple geometric principles, "the radius of curvature of an equiangular spiral is proportional to the arc s, measured from the origin. In fact, $$ \rho=s \cot \phi . $$