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Precalculus MM: Lair of the Devourer: Vectors and Parametric Equations 3: NWN 0140c

The Stone of Recall also uses the vector equation of a line to calculate the velocity and speed of Zanzi's movement if she moves at a constant velocity (if her velocity is constant then by definition she must be moving in a straight line). In this instance the direction vector v is the velocity vector. In order to de-model Zanzi's constant motion along a straight line, occasionally the Stone of Recall will deconstruct the vector equation < x, y > = < x subzero, y subzero > + t< x subone, y subone > to obtain two equations, one in terms of the x components and t and the other in terms of the y components and t, i.e. x = x subzero +t(x subone) and y = y subzero + t(y subone). These two equations in terms of x and t and y and t are called parametric equations, and sometimes the Stone of Recall will eliminate the t parameter in order to obtain rectangular equations that facilitate graphing Zanzi's motion, particularly when using more complex parametric equations used to de-model her non-linear movements.
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De-modelling Zanzi's motion to aid graphing her movements means that the Stone of Recall will parameterise vector equations of line segments of her motion by setting t to zero to represent the beginning of her motion to de-model and 1 to represent the end of the motion to de-model, so 0<= t <=1. Within this domain restriction on scalar multiplier t, the Stone of Recall will commence de-modelling her movements from point A, the starting point when t=0, and end the de-modelling at point B when t=1. Any parallel vector point P is used to in the de-modelling process when the scalar multiplier t is at some value of interest between t = 0 and t = 1. After obtaining the component values for the coordinates of the fractional vector line segment AP and full-length vector line segment AB, the Stone of Recall will parameterise the vector equation (e.g. for AP = tAB < x - x subone, y - y subone> = t< x subtwo - x subone, y subtwo - y subone >) by separating the components. This eventually yields x = x subone (1 - t) + tx subtwo and y = y subone (1 - t) + ty subtwo. Values of t within its domain restriction will now provide coordinates values for x and y for any point along the line segment of motion.
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In a procedure known the the arc length parameterisation, where the scalar multiplier when parameterising a line segment is a distance (usually denoted as "s") rather than a time, t, the Stone of Recall will first use PT to find the total length of the line segment vector once the segment's components have been computed from the beginning and end point coordinates. It will then calculate a unit direction vector, v, (i.e. length one unit) and set it parallel to (i.e. overlapping) the line segment vector. It will then use the distance scalar multiplier, s, on the unit direction vector, v, and set that product equal to vector AP from starting point A of the line segment, such that vector AP = sv. Parameterisation of this yields the x and y equations in terms of the s parameter. As usual when parameterising a line segment, the scalar multiplier will have a domain restriction to keep the x and y coordinates for all points calculated using the parametric equations within the end points of the line segment.
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