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3 years ago

Please help on 13 and 14

Mathematics

1 answer:

alexira [117]3 years ago

4 0

Answer:

13) The planes ABF and FHG are perpendicular

14) Segments Ab and CD are parallel.

Step-by-step explanation:

13)

Planes ABF and FHG are perpendicular (they intersect along side EF)

14)

In order to find out about the segments AB and CD, let's use the formula for the slope that joins any two points $(x_1,y_1)\,\,\&\,\,(x_2,y_2)$ on the plane:

$slope=\frac{y_2-y_1}{x_2-x_1}$

Therefore, given:

A = (-3, 2)

B = (1, 4)

C = (-3, -1)

D = (1, 1)

we have for segment AB the following slope:

$slope_{AB}=\frac{4-2}{1+3} =\frac{2}{4} =\frac{1}{2}$

and for segment CD the following slope:

$slope_{CD}=\frac{1+1}{1+3} =\frac{2}{4} =\frac{1}{2}$

Since they show the same slope, these two segments are parallel.

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3 years ago

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2 years ago

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3 years ago

Find the arc-length parametrization of the curve that is the intersection of the elliptic cylinder x 2 + y 2/2 = 1 and the plane

storchak [24]

Answer:

$f(\theta) = (cos(\frac{\theta}{\sqrt2}), \sqrt2 sin(\frac{\theta}{\sqrt2}), cos(\frac{\theta}{\sqrt2})-2)$

0 ≤ Ф ≤ 4π.

Step-by-step explanation:

since x²+y²/2 = 1, then x²+s² = 1, with s = (y/√2)². Hence, (x,s) = (cos(Ф),sin(Ф)) and (x,y,z) = (cos(Ф),√2 sin(Ф), cos(Ф)-2). This expression evaluated in zero gives as result (1,0,-1). The derivate of this function is (-sin(Ф),√2 cos(Ф), -sen(Ф))

the norm of the derivate is √(sin²(Ф) + 2cos²(Ф)+sin²(Ф)) = √2. In order to make the norm equal to 1, i will divide Ф by √2, so that a √2 is dividing each term after derivating.

We take

$f(\theta) = (cos(\frac{\theta}{\sqrt2}), \sqrt2 sin(\frac{\theta}{\sqrt2}), cos(\frac{\theta}{\sqrt2})-2)$

Note that

$f(0) = (1,0,-1)$
$f'(\theta) = (\frac{sin(\frac{\theta}{\sqrt2})}{\sqrt2}, cos(\frac{\theta}{\sqrt2})}, \frac{sin(\frac{\theta}{\sqrt2})}{\sqrt2})$

Whose square norm is 1/2cos²(Ф/2)+sen²(Ф/2)+1/2cos²(Ф/2) = 1. This is te parametrization that we wanted.

The values from Ф range between 0 an 4π, because the argument of the sin and cos is Ф/2, not Ф, Ф/2 should range between 0 and 2π.

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3 years ago