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Alona [7]
3 years ago
14

Consider a sphere of radius R > 0. Take two antipodal points A and B on the sphere, and another point C on the sphere. Check

that the angle ACB is a right angle.
Use only vectors. No solution involving classical geometry or coordinates will be accepted.

You may want to draw a picture, and introduce a good coordinate system.

Mathematics
1 answer:
aleksklad [387]3 years ago
5 0

Answer:

To this question; we want to show that if two antipodal points on a sphere were A and B, for any random point C on the sphere, AC is perpendicular to BC?

Step-by-step explanation:

I would proceed by imagining the sphere in a three dimensional Cartesian coordinate system. For example, use a sphere of diameter 2 and let it sit at the origin. Then (0, 0, 1) and (0, 0, -1) are the vector locations of the “north and south poles” of the sphere.

Now choose the vector location of any point on the surface of the sphere. It will have a vector location - we’ll call it (x, y, z). Now the vectors from your point to the two poles are (-x, -y, 1-z) and (-x, -y, -1-z).

Now just form the dot product of those two vectors:

(-x, -y, 1-z) . (-x, -y, -1-z) = x^2 +y^2 + (1-z)*(-1-z)

Now the truth of your claim will be embodied in that dot product being zero:

x^2 + y^2 - (1+z)(1-z) = 0

x^2 + y^2 - (1-z^2) = 0

x^2 + y^2 + z^2 = 1

But that last line is just the definition of points on the surface of a sphere of radius 1, so the claim is proven.

Since R>0 and AC is perpendicular to BC, the <ACB is at right angle.

Please, find attached a simple image to show antipodal points (Two points that makes a diameter) and a point C on the sphere.

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Twice x, plus 7, is the same as – 10.<br> The answer is N
sergij07 [2.7K]

Answer:

x=-8.5.

Step-by-step explanation:

First, write an equation. Twice x (2x) plus 7 (+7) is the same as (=) negative ten (-10), or 2x+7=-10. To solve, first subtract 7 from both sides to get 2x=-17. Then, divide both sides by 2 (x=-8.5).

3 0
3 years ago
Solve log 4x + log 13 = 0. Round to the nearest thousandth if necessary. a.0.019 b. 52 c.3.25 d. 0.311 Mathematics
kumpel [21]
Log4 + logX + log13 = 0
logX = -log13 - log4
logX = -1.716
x= 10^[-1.716]
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6 0
3 years ago
Joe can cut and split a cord of firewood in 3 fewer hours than Dwight can. When they work​ together, it takes them 2 hours. How
aalyn [17]

Answer:

Dwight will take 6 hours to finish the job alone and Joe will take 3 hours to finish the job alone.

Step-by-step explanation:

Let us assume the time taken by Dwight to split a cord of firewood  = K hrs

So, the per hour rate of Dwight  = (\frac{1}{K})

As, Joe uses 3 LESS hours then Dwight.

So, the time taken by Joe to split a cord of firewood  = (K- 3) hrs

So, the per hour rate of Joe  = (\frac{1}{K-3})

Now, when both of them wok together, it takes them 2 hours.

So, the per hour rate of BOTH of them  = (\frac{1}{2})

⇒ Per hour rate of ( Dwight  + Joe)  =(\frac{1}{2} )

\implies (\frac{1}{K}) +  (\frac{1}{K-3}) = (\frac{1}{2})

Now, solving for the value of K , we get:

(\frac{1}{K}) +  (\frac{1}{K-3}) = (\frac{1}{2})\\\implies  \frac{(K-3) + K}{K (K-3)}  = (\frac{1}{2})\\\implies 2(2K -3) = K^2 - 3K\\\implies k^2 - 3K -4K +6 = 0\\\implies K^2 - 7K  + 6=  0\\\implies K^2 - 6K - K  + 6=  0\\\implies K(K-6) -1(K - 6)=  0\\\implies (K-6)(K-1) = 0

Implies either K = 6 Or K = 1

But if K = 1, (K-3)  = 1- 3  = -2 hours would be A CONTRADICTION.

⇒ K  = 6 hours

Hence, Dwight will take 6 hours to finish the job alone and Joe will take (k-3) = (6-3) = 3 hours to finish the job alone.

8 0
3 years ago
What is the distance between the points (2,10) and (-6, 4) on the coordinate
stellarik [79]

Answer:

The distance between the given points (2,10) and (-6, 4) on the coordinate  plane is 10units

Therefore distance s=10 units

Step-by-step explanation:

Given points are (2,10) and (-6, 4) on the coordinate plane

To distance between the given points :

The distance formula is s=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2} units

Let (x_1,y_1) ,(x_2,y_2) be the given points (2,10) and (-6, 4) respectively

Now substituting the values in the distance formula  we get

s=\sqrt{(-6-2)^2+(4-10)^2}

=\sqrt{(-8)^2+(-6)^2}

=\sqrt{8^2+6^2}

=\sqrt{64+36}

=\sqrt{100}

=10

Therefore s=10 units

The distance between the given points (2,10) and (-6, 4) on the coordinate plane is 10units

5 0
3 years ago
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Leokris [45]
Choice given:
<span>33.6°
39.8°
50.2°
56.4°

I drew the figure. 
I got 12 ft as the hypotenuse, 10 ft as the opposite.

Sin</span>Θ = opposite / hypotenuse 
SinΘ = 10/12
SinΘ = 0.83

I used the calculator to get the value of each angle using the sine function
sin(33.6°) = 0.55
sin(39.8°) = 0.64
sin(50.2°) = 0.77
sin(56.4°) = 0.83

The angle where the wire meets the ground is approximately 56.4°
8 0
3 years ago
Read 2 more answers
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