Thus, the camera angle remains objective, since it is an unseen observer not involved in the action.

Thus the spinor rotation γ ( ψ ) = γψ ( ordinary quaternionic multiplication ) will rotate the spinor ψ through an angle one-half the measure of the angle of the corresponding vector rotation.

where θ is the angle between the vectors Ω < sub > P </ sub > and L. Thus, if the gyroscope's spin slows down ( for example, due to friction ), its angular momentum decreases and so the rate of precession increases.

Thus, a control-canard mostly operates only as a control surface and is usually at zero angle of attack.

Thus, the object's hour angle indicates how much sidereal time has passed since the object was on the local meridian.

Thus, the true anomaly is shown as the red angle in the diagram, and the mean anomaly is not shown.

Thus the sine law is equivalent to the angle sum identity when the angles sum is between 0 and 180 degrees and when each individual angle is between 0 and 90 degrees.

Thus compass and straightedge geometry solves second-degree equations, while origami geometry, or origametry, can solve third-degree equations, and solve problems such as angle trisection and doubling of the cube.

Thus the length of a tangent vector cannot be defined, but the angle between two vectors still can.

Thus the hyperbolic angle and the hyperbolic functions sinh, cosh, and tanh are all transcendental.

Thus, it is the ratio between height and transition radius that determines the personality of a given ramp, because the ratio determines the angle of the lip.

Thus the polarization undergoes parallel transport and the phase shift is given by the enclosed solid angle ( times the spin, which in case of light is 1 ).

Thus, a dihedral angle can be defined by four, pairwise non-collinear vectors.

Thus, an angle measuring 90 degrees is called a right angle.

Thus the colors produced vary with viewing angle, however they are actually surprisingly uniform, perhaps due to the tetrahedral ( diamond-like ) structural arrangement of the scales or diffraction from overlying cell layers.

Thus the supplement of an angle of x degrees is an angle of ( 180 − x ) degrees.

Thus, the Four Thirds System has crop factor ( focal length multiplier ) of about 2, and while this enables longer focal length for greater magnification, it doesn't aid the manufacture of wide angle lenses.

Thus the player can be presented an overview of the game world in the ability to see it from above, more or less, and with additional details in artwork made possible by using an angle: Instead of showing a humanoid in top-down perspective, as a head and shoulders seen from above, the entire body can be drawn when using a slanted angle ; Turning a character around would reveal how it looks from the sides, the front and the back, while the top-down perspective will display the same head and shoulders regardless.

Thus from the military angle the same mistake is being made as in 1918-only more senselessly ; one has not the courage to face the inevitable facts in time.

Thus, u < sub > ρ </ sub > and u < sub > θ </ sub > form a local Cartesian coordinate system attached to the particle, and tied to the path traveled by the particle.

Thus, when light is normally incident on the grating, the diffracted light will have maxima at angles θ < sub > m </ sub > given by:

Thus, psi ( ψ ) may be written as ps, 4 or y ; xi ( ξ ) as ks, x or 3 ; and theta ( θ ) as th or 8.

Thus, if θ is one value of arg ( z ), the other values are given by arg ( z ) = θ + 2nπ, where n is any integer ≠ 0.

The function f is computable, so there is a formula δ representing f in T. Thus for each formula θ, T proves

Thus, a vacuum with ⟨ θ ⟩= 0 will shift to a different vacuum with ⟨ θ ⟩=− ε.

Thus vector ( x, y, z ) is the unit axis of rotation scaled by tan < sup > θ </ sup >⁄< sub > 2 </ sub >.

Thus, it is a special case of a phase shift gate ( next ) with θ = π.

Thus for either of the similar right triangles in the figure, the ratio of its horizontal side to its hypotenuse is the same, namely cos θ.

Thus we have a formula for the rotation matrix R corresponding to an axis angle representation θ:

Thus the maximum distance is obtained if < var > θ </ var > is 45 degrees.

Thus, when given a Poisson regression model θ and an input vector, the predicted mean of the associated Poisson distribution is given by

Thus Af is also continuous at Af, and in a neighborhood of Af which does not contain a tangent point.

Thus, in the immemorial way -- in the way of the right hand that knows and the left that does not -- was the stage set for tragedy at Bari.

The evaluation order does not affect the value of such expressions, and it can be shown that the same holds for expressions containing any number of operations .< ref > Thus, when is associative, the evaluation order can be left unspecified without causing ambiguity, by omitting the parentheses and writing simply:

Thus, when the fluid finally does strike the object, it is forced to change its properties -- temperature, density, pressure, and Mach number -- in an extremely violent and irreversible fashion called a shock wave.

Thus, the Big Bang theory cannot and does not provide any explanation for such an initial condition ; rather, it describes and explains the general evolution of the universe going forward from that point on.

Thus, while the book exhibits considerable unity and probably reflects much of the historic Ezekiel, it is the product of a long and complex history and does not necessarily preserve the very words of the prophet.

Thus, the Buddha's silence does not indicate misology or disdain for philosophy.

Thus, one nucleus offers a more tightly bound position to an electron than does another nucleus, with the result that one atom may transfer an electron to the other.

Thus, covalent bonding does not necessarily require the two atoms be of the same elements, only that they be of comparable electronegativity.

Thus, though often de facto the case, Conservative Judaism's halakhic system does not inherently see Orthodox halakhic practice as acceptable and legitimate halakhic practice for a Conservative Jew.

Thus query optimization typically tries to approximate the optimum by comparing several common-sense alternatives to provide in a reasonable time a " good enough " plan which typically does not deviate much from the best possible result.

Thus, according to Lutz, doublespeak is a form of language that defeats the purpose of inventing language because doublespeak does not communicate the truth but seeks to do the opposite and the doublespeak committee is tasked with correcting this problem that doublespeak has created in the world of language.

Thus query optimization typically tries to approximate the optimum by comparing several common-sense alternatives to provide in a reasonable time a " good enough " plan which typically does not deviate much from the best possible result.

Thus, interlaced video captures samples the scene motion twice as often as progressive video does, for the same number of frames per second.

Thus, at UV frequencies and higher ( and probably somewhat also in the visible range ), electromagnetic radiation does far more damage to biological systems than simple heating predicts.

Thus, according to Hegel, a great man does not create historical reality himself but only uncovers the inevitable future.

Thus, Annie's strong drive to take revenge on women scholars in general and on Miss de Vine in particular is perfectly comprehensible, and in fact does not in itself prove her to be mentally deranged.

Thus the group of left and right handed 4x4 quaternion matrcies is Sp ( 8 ) xSU ( 2 ) which does include the standard model bosons:

Thus, " DD " does not stand for anything more than " Destroyer ".

Thus it does not stick to the neck and scalp when wet.

Thus he does not scruple to declare that Socrates and Heraclitus were Christians ( Apol., i. 46, ii.

Thus, union density provides a rough picture of union membership only ; it does not account for the circumstance that in some countries, also many persons under education, many unemployed persons, many retired persons and / or many persons who had to leave work due to occupational injuries may also be union members.

Thus in 1992, the Catechism of the Catholic Church, while affirming that " the Church does not know of any means other than Baptism that assures entry into eternal beatitude ", but also stating that " God has bound salvation to the sacrament of Baptism, but he himself is not bound by his sacraments ", stated: " As regards children who have died without Baptism, the Church can only entrust them to the mercy of God, as she does in her funeral rites for them.

Thus, Merleau-Ponty does not postulate that " all consciousness is consciousness of something ", which supposes at the outset a noetic-noematic ground.