In the Euclidean plane, the angle θ between two vectors u and v is related to their dot product and their lengths by the formula

On the plane the most common alternative is polar coordinates, where every point is represented by its radius r from the origin and its angle θ.

where θ is the true angle SEE ′, is the apparent angle S ′ EE ′, and is the relative speed between the presumed fixed frame of reference ( such as heliocentric ) and the observer's one.

Using F for the drag force, ρ for the density, S for the area of the flat plate, V for the flow velocity, and θ for the inclination angle, his law was expressed as

where θ < sub > r, p </ sub > is the angle between r and p measured from r to p ; an important distinction because without it, the sign of the cross product would be meaningless.

The curve is banked at an angle θ from the horizontal, and the surface of the road is considered to be slippery.

As the angle of bank θ approaches 90 °, the tangent function approaches infinity, allowing larger values for | v |< sup > 2 </ sup >/ R.

When the angle θ does not satisfy the above condition, the horizontal component of force exerted by the road does not provide the correct centripetal force, and an additional frictional force tangential to the road surface is called upon to provide the difference.

The direction of u < sub > r </ sub > is described by θ, the angle between the x-axis and the unit vector, measured counterclockwise from the x-axis.

By moving the unit vectors so their tails coincide, as seen in the circle at the left of the image above, it is seen that u < sub > ρ </ sub > and u < sub > θ </ sub > form a right-angled pair with tips on the unit circle that trace back and forth on the perimeter of this circle with the same angle θ ( t ) as r ( t ).

Using the tangent vector, the angle θ of the tangent to the curve is given by:

The path difference is given by so that the minimum intensity occurs at an angle θ < sub > min </ sub > given by

* θ < sub > i </ sub > is the angle at which the light is incident,

: θ is the angle of the diffracted wave.

The ( non-reflex ) angle θ ( 0 ° ≤ θ ≤ 180 °) between x and y is then given by

So the parameters are: a — distance from center C to either vertex b — length of a perpendicular segment from each vertex to the asymptotes c — distance from center C to either Focus point, F < sub > 1 </ sub > and F < sub > 2 </ sub >, and θ — angle formed by each asymptote with the transverse axis.

Consistent with the symmetry of the hyperbola, if the transverse axis is aligned with the x-axis of a Cartesian coordinate system, the slopes of the asymptotes are equal in magnitude but opposite in sign, ±, where b = a × tan ( θ ) and where θ is the angle between the transverse axis and either asymptote.

In 1805, Thomas Young defined the contact angle θ by analyzing the forces acting on a fluid droplet resting on a solid surface surrounded by a gas.

The contact angle, θ < sub > C </ sub >, is the angle formed by a liquid at the three-phase boundary where the liquid, gas, and solid intersect.

A droplet resting on a solid surface and surrounded by a gas forms a characteristic contact angle θ.

Lowering a concentrated, heavy weight down the anchor line – rope or chain – directly in front of the bow to the seabed, behaves like a heavy chain rode and lowers the angle of pull on the anchor.

Similarly, the angle that a line makes with the horizontal can be defined by the formula

For the simple case where the direction of the light beam along the line SE is perpendicular to the motion of the observer then only light emitted at an angle equal to the arc cosine of the ratio of the observer's speed to the speed of light will reach the telescope.

In addition, a rotation axis at an angle to the magnetic axis may be spinning different bands of magnetically-sorted elements into the line of sight between Alioth and the Earth.

Fishing line lends itself better to right angle weave because it is stiffer than nylon thread, so holds the beads in a tighter arrangement and does not easily break when tugged upon.

Right angle weave is done using both ends of the fishing line, in which beads are strung in repeated circular arrangements, and the fishing line is pulled tight after each bead circle is made.

In more generality, reflection across a line through the origin making an angle with the x-axis, is equivalent to replacing every point with coordinates ( x, y ) by the point with coordinates ( x < nowiki >'</ nowiki >, y < nowiki >'</ nowiki >), where

* Disclinations are line defects corresponding to " adding " or " subtracting " an angle around a line.

The eccentric anomaly t is the angle of the blue line with the X-axis.

Thales ' theorem, named after Thales of Miletus states that if A, B, and C are points on a circle where the line AC is a diameter of the circle, then the angle ABC is a right angle.

Euler discussed a generalization of Euclidean geometry called affine geometry, which retains the fifth postulate unmodified while weakening postulates three and four in a way that eliminates the notions of angle ( whence right triangles become meaningless ) and of equality of length of line segments in general ( whence circles become meaningless ) while retaining the notions of parallelism as an equivalence relation between lines, and equality of length of parallel line segments ( so line segments continue to have a midpoint ).

Here, x is the angle that a line connecting the origin with a point on the unit circle makes with the positive real axis, measured counter clockwise and in radians.

Which conic section is formed depends on the angle the plane makes with the axis of the cone, compared with the angle a line on the surface of the cone makes with the axis of the cone.

If the angle between the plane and the axis is less than the angle between the line on the cone and the axis, or if the plane is parallel to the axis, then the conic is a hyperbola.

Experiments showed that the surface chemistry and geometry at the contact line affected the contact angle and contact angle hysteresis, but the surface area inside the contact line had no effect.

The various forces result from the reaction of the removed paint chip against the face of the knife and along the shear plane, which makes an angle **yf with the substrate.

**yf is the angle the shear plane makes with the substrate ; ;

Rail fins are typically " toed-in ," that is, the leading edge of the fins are oriented toward the centerline of the surfboard, which decreases the angle of attack in trim, which makes it easier to initiate turns.

Nutation makes a small change to the angle at which the Earth tilts with respect to the Sun, thereby moving the location of the Tropic of Cancer, the most northerly latitude which the Sun can reach directly overhead.

In specular reflection, the direction of the reflected ray is determined by the angle the incident ray makes with the surface normal, a line perpendicular to the surface at the point where the ray hits.

It is an angular velocity about a line that makes an angle with the permanent rotation axis, and this angle lies in a plane at right angles to the plane of the couple producing the torque.

The seasons occur because the Earth's axis of rotation is not perpendicular to its orbital plane ( the “ plane of the ecliptic ”) but currently makes an angle of about 23. 44 ° ( called the " obliquity of the ecliptic "), and because the axis keeps its orientation with respect to an inertial frame of reference.

The path of the light to a position on the screen is different for the two slits, and depends upon the angle θ the path makes with the screen.

These principles include the bare basics of what makes objects structurally sound and, by extension, beautiful ( i. e. the angle and the fixed / static shapes that it creates — esp.

Additionally, a line drawn through the femoral neck superimposed on a line drawn through the femoral condyles forms an angle, the torsion angle, which makes it possible for flexion movements of the hip joint to be transposed into rotary movements of the femoral head.

It gets its name from its operating principle, the tangent law of magnetism, which states that the tangent of the angle a compass needle makes is proportional to the ratio of the strengths of the two perpendicular magnetic fields.

For step index fiber, light entering the fiber will be guided if it falls within the acceptance cone of the fiber, that is if it makes an angle with the fiber axis that is less than the acceptance angle,

: θ is the angle the ray makes with the fiber axis, before entering the fiber,

This makes Polaris useful for navigation in the northern hemisphere: not only is it always above the north point of the horizon, but its altitude angle is always ( nearly ) equal to the observer's geographic latitude.

Resin makes a bowling ball tackier than it would otherwise be, increasing its ability to hook into the pins at an angle and ( with correct technique ) making strikes easier to achieve.