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Yuri [45]
2 years ago
15

Find the equation of the plane through the point (2,5,7) that is parallel to the line r=(3i+2j−2k)+t(i+2j+9k) and perpendicular

to the plane 4x+5y+6z=14. Write the equation in the form indicated.
Mathematics
1 answer:
insens350 [35]2 years ago
7 0

The plane we want to find has general equation

a(x-2)+b(y-5)+c(z-7)=0

with a,b,c not equal to 0, and has normal vector

\vec n=a\,\vec\imath+b\,\vec\jmath+c\,\vec k

\vec n is perpendicular to both the normal vector of the other plane, which is 4\,\vec\imath+5\,\vec\jmath+6\,\vec k, as well as the tangent vector to the line \vec r(t), which is \vec\imath+2\,\vec\jmath+9\,\vec k.

This means the dot product of \vec n with either vector is 0, giving us

\begin{cases}4a+5b+6c=0\\a+2b+9c=0\end{cases}

Suppose we fix c=1. Then the system reduces to

\begin{cases}4a+5b=-6\\a+2b=-9\end{cases}

and we get

(4a+5b)-4(a+2b)=-6-4(-9)\implies-3b=30\implies b=-10

a+2(-10)=-9\implies a=11

Then one equation for the plane could be

\boxed{11(x-2)-10(y-5)+(z-7)=0}

or in standard form,

\boxed{11x-10y+z=-21}

The solution is unique up to non-zero scalar multiplication, which is to say that any equation (11x-10y+z)k=-21k would be a valid answer. For example, suppose we instead let c=2; then we would have found a=22 and b=-20, but clearly dividing both sides of the equation

22(x-2)-20(y-5)+2(z-7)=0

by 2 gives the same equation as before.

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Can someone help me with this
Anvisha [2.4K]

Answer:

D. \frac{{7}^{11} }{ {4}^{11} }

Step-by-step explanation:

( \frac{7}{4}) {}^{11} = \frac{7 {}^{11} }{4 {}^{11} }

Since we are multiplying an exponential number to a fraction the answer is going to be

\frac{7 {}^{11} }{4 {}^{11} }

Hope this helps ❤❤❤ ;)

7 0
2 years ago
Read 2 more answers
Demarco and Tanya have received information about three separate mortgage offers. In two or three paragraphs, describe your reco
Mars2501 [29]

Answer:

Mortgage option (3) would be best suited for them.

Step-by-step explanation:

Mortgage option (1) and (2) are more or less the same since, since even if Damarco and Tanya down payments $34,000 (20% of the purchase price), they need to pay the interest for 30 years for both of the cases and even if he pays  about $750   monthly (as for option (1)) or about  $ 9000 annually (as  for option (2))  both may actually be more or less the same amount  since, the annual rate of interest in (2) may increase from the initial rate of 3.5% (but it is very unlikely to increase to over 5%) and option (1) has an annual  fixed rate of interest of 4.25%.

Now, in the option (3) the interest is to be paid for 8 years and the annual rate of interest is also relatively low (only 4%) and if they pay about $18,000 annually with a down-payment of $ 34,000 and  repay the rest of the amount at the end of 8 years,(which would be less than $ 35,000)  they can easily clear their mortgage. Hence, for option (3) they would need to pay lowest total amount and for lowest time to clear the mortgage among the three options. Hence, this would be best suited option for them.

6 0
2 years ago
A teacher reviews 4-1/2 papers per hour, how many papers will that teacher review in 6-1/3 hours
MrRa [10]

Answer:

28\frac{1}{2}\ papers

Step-by-step explanation:

step 1

Convert mixed numbers to an improper fractions

4\frac{1}{2}=\frac{4*2+1}{2}=\frac{9}{2}

6\frac{1}{3}=\frac{6*3+1}{3}=\frac{19}{3}

step 2

we know that

A teacher reviews 9/2 papers per hour

using proportion

Find how many papers will the teacher review in 19/3 hours

Let

x-----> the number of papers

(9/2)/1=x/(19/3)

x=(19/3)(9/2)

x=28.5\ papers

Convert to mixed numbers

28.5=28\frac{1}{2}\ papers

5 0
2 years ago
Add. ˉ8 + ˉ6 = ____
Nutka1998 [239]
The answer is D +14 because to negatives make a positive
3 0
2 years ago
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Box 1: Dimensions: x by 3x by x3 Area of base = x(3x) = 3x2 Box 2: Dimensions: x by 4x – 1 by x3 Area of base = x(4x – 1) = 4x2
andriy [413]

Answer:

box 1: monomial

box 2: binomial

Step-by-step explanation:

reasoning: Box 1’s volume is modeled by a monomial times a monomial, so it will be a monomial.

Box 2’s volume is modeled by a monomial times a binomial, so it will be a binomial.

4 0
2 years ago
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