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3 years ago

Find an equation of the plane consisting of all points that are equidistant from (-1, 1, -5) and (4, -5, 0).

Mathematics

1 answer:

Leno4ka [110]3 years ago

8 0

Answer: the required equation is 5x - 6y + 5z = 7

Step-by-step explanation:

Given that;

points : (-1, 1, -5) and (4, -5, 0).

mid point : ( [(-1+4)/2], [(1-5)/2], [(-5+0)/2]

⇒ midpoint : ( 3/2, -2, -5/2 )

x₀ y₀ z₀

Direction vector n = [4-(-1)], [ -5 - 1], [ 0 - (-5)]

⇒ Direction vector n = < 5, -6, 5 >

General equation plane : n(x-x₀, y-y₀, z-z₀) = 0

so we substitute

⇒ (5, -6, 5) (x-3/2, y-(-2), z-(-5/2) ) = 0

⇒ (5, -6, 5) (x - 3/2, y + 2, z + 5/2) ) = 0

⇒ 5(x - 3/2) - 6(y + 2 ) + 5(z + 5/2) = 0

⇒ 5x - 15/2 - 6y - 12 + 5z + 25/2 = 0

⇒ 5x - 6y + 5z = 15/2 + 12 - 25/2

⇒ 5x - 6y + 5z = 7

Therefore, the required equation is 5x - 6y + 5z = 7

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pychu [463]

I think you mean to say $\cos^2\theta=\dfrac14$ . If $\theta$ is in the third quadrant, then $\sin\theta$ and $\cos\theta$ . Recall that

$\sin^2\theta+\cos^2\theta=1\implies\sin\theta=\pm\sqrt{1-\cos^2\theta}$

We expect $\sin\theta$ to be negative, so we take the negative root. We end up with

$\sin\theta=-\sqrt{1-\dfrac14}=-\dfrac{\sqrt3}2$

We also know to expect $\cos\theta$ , so

$\cos\theta=\pm\sqrt{\dfrac14}\implies\cos\theta=-\dfrac12$

By definition, we have

$\cot\theta=\dfrac{\cos\theta}{\sin\theta}$

and so

$\cot\theta=\dfrac{-\frac12}{-\frac{\sqrt3}2}=\dfrac1{\sqrt3}$

7 0

3 years ago

What is the area of this figure?

KatRina [158]

Hi!

I have attached 2 images that should help you understand :)

First, look at the edits I made to the image you posted. I separated the shape into smaller shapes so that we can find the area of each individual one.

Let's start with the rectangle.

To find the area of a rectangle, multiply the width times the height.
10 · 4 = 40

Rectangle = 40cm

Next up, the red triangles.

I have included another image showing the triangles combined into rectangles. So we can find the area of the triangles just like we would rectangles!

(let me know if you don't understand how I found the width + height of the triangles)

5 · 10 = 50

Red triangles = 50cm

And finally, the green triangles.

8 · 7 = 56

Green triangles = 56cm

Add it all together and you get...

40 + 50 + 56 = 146

The answer to the question is 146cm.

Next time you are having trouble with something like this, picture the triangles as rectangles! :)

3 0

3 years ago

95. Because atoms, molecules, and ions are very small, they are counted in units of moles, where 1 mole = 6.022 × 1023 molecules

Galina-37 [17]

Answer:

There are 10⁻⁷ hydrogen ions in one-liter water

Step-by-step explanation:

Hi there!

a. If the pH = 7.0, then:

-log(H) = 7.0

multiply both sides of the equation by -1

log(H) = -7.0

Apply 10ˣ to both sides

10^(log (H)) = 10⁻⁷

(H) = 10⁻⁷

There are 10⁻⁷ hydrogen ions in one-liter water.

Why 10^(log (H)) = (H)?

Let´s consider this function:

y = 10^(log x)

(Apply log to both sides)

log y = log 10^(log x)

(Apply logarithmic property: log xᵃ = a · log x)

log y = log x · log 10

(log 10 = 1)

log y = log x

y = x

(since y = 10^(log x) and y = x):

10^(log x) = x

That´s why 10^(log (H)) = (H)

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3 years ago

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ella [17]

Answer:

Step-by-step explanation:

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3 years ago

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kiruha [24]

Answer:

Step-by-step explanation:

Believe it or not, the two areas are the same.

The base of the rectangle is PQ

The height of the rectangle is PS

Now look at the parallelogram.

The base is PQ

The height is PS

The area has to be the same in both cases. There is no other way to interpret what is happening.

4 0

3 years ago