Question

Sketch the curve represented by the parametric equations. Use arrows to indicate the direction of the curve as t increases. Find a rectangular-coordinate equation for the curve by eliminating the parameter. Express the vector v with initial point P and terminal point Q in component form. (Assume that each point lies on the gridlines.)

Answer 1

The lower right image represents the image of the parametric formulas, whose rectangular formula is $\frac{x^{2}}{16} + \frac{y^{2}}{25} = 1$. The vector in component form is $\vec v = (8, -4)$.

How to analyze parametric equations and vectors

Parametric formulas are vectorial expressions in terms of a parameter (t). Planar parametric expression are of the form $\vec r(t) = (x(t), y(t))$. Ellipses centered at the origin are described by the following expression:

$\vec r (t) = (a\cdot \cos t, b \cdot \sin t)$     (1)

Where a, b are the lengths of the major and minor semiaxes.

By direct observation of the given parametric equations, we conclude that the ellipse of the lower right image represents the two equations.

The rectangular equation of the ellipse is found by eliminating the parameter:

$\cos ^{2}t + \sin ^{2}t = 1$  

$(\frac{x}{4})^{2} + \left(\frac{y}{5} \right)^{2} = 1$

$\frac{x^{2}}{16} + \frac{y^{2}}{25} = 1$

According to the geometry, vectors can be generated from two points, one of them as the initial point. A vector can be defined as a subtraction between two vectors with initial points at the origin:

$\vec v = B(x, y) - A(x, y)$     (2)

Where:

• A(x, y) - Initial point
• B(x, y) - Final point

If we know that A(x, y) = (1, 8) and B(x, y) = (9, 4), then the equation of the vector is:

$\vec v = (9, 4) - (1, 8)$

$\vec v = (9 - 1, 4 - 8)$

$\vec v = (8, -4)$

To learn more on parametric equations: https://brainly.com/question/12718642

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