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A sailboat race course consists of four legs, defined by the displacement vectors A, B, C, and D as the drawing indicates. The magnitudes of the first three vectors are A=6.4 km , B=5.10 km, and C=4.80 km. The finish line of the course coincides with the starting line. Using the data in the drawing, find the distance of the fourth leg and the angle theta.

A sailboat race course consists of four legs, defined by the displacement vectors A, B, C, and D as the drawing indicates. The magnitudes of the first three vec class=

Answer :

MissPhiladelphia
The first step is to solve the horizontal component using the given values from the problem and the illustration,
horizontal component = 6.4sin(40°) + 5.1sin(35°) - 4.8sin(23°)

Then the vertical component is determined,
vertical component = 6.4cos(40°) - 5.1cos(35°) - 4.8cos(23°)

The distance of the fourth leg is calculated using the Pythagorean Theorem,
distance = (h²+v²)^1/2 = 6.35 km

Finally, the angle is computed by applying the inverse tangent function,
theta = tan^-1(v/-h) = 54.42°
xero099

The distance of the fourth leg is approximately 7.145 kilometers and the angle is approximately 58.879°.

Determinación de la magnitud y el ángulo de un vector a partir de una suma de vectores

El vector [tex]D[/tex] es la suma de los tres vectores [tex]A[/tex], [tex]B[/tex] y [tex]C[/tex], cuya operación se ve a continuación:

[tex]\vec D = -\vec A - \vec B - \vec C[/tex]   (1)

[tex]\vec D = -(6.4\cdot \cos 40^{\circ}, 6.4\cdot \sin 40^{\circ}) - (-5.10\cdot \cos 35^{\circ}, 5.10\cdot \sin 25^{\circ}) - (-4.80\cdot \cos 23^{\circ}, -4.80\cdot \sin 23^{\circ})[/tex]

[tex]\vec D = (3.693, -6.117)[/tex]

The magnitude is determined by Pythagorean theorem:

[tex]D = \sqrt{3.693^{2}+(-6.117)^{2}}[/tex]

[tex]D \approx 7.145\,km[/tex]

The direction is found by this inverse trigonometric relationship:

[tex]\theta_{D} = \tan^{-1} \frac{6.117\,km}{3.693\,km}[/tex]

[tex]\theta_{D} \approx 58.879^{\circ}[/tex]

The distance of the fourth leg is approximately 7.145 kilometers and the angle is approximately 58.879°.

To learn more on vectors, we kindly invite to check this verified question: https://brainly.com/question/13188123

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