Find the circumference of the circle with radius r cm

Find a unit vector orthogonal to the plane containing the points A= 1,0,0 , B= 3,−1,−3 , and C= 1,3,−2 .

zysi [14]

Answer:

unit normal vector n will be n=(a,b,c) = (4/√171,11/√171,6/√171)

Step-by-step explanation:

There are several ways to solve this problem

1) build 2 vectors AB and BC such that the vectorial product ABxBC is the orthogonal vector to the plane , then find unit vector

2) since the 3 points belongs to the plane solve a linear system of 4 equation with 4 variables

for the second solution , the equation of the plane with normal vector n=(a,b,c) and containing the point (x₀,y₀,z₀) is

a*(x-x₀)+b*(y-y₀)+c*(z-z₀) =0

and

a²+b²+c² = 1 (unit vector)

then choosing A=(x₀,y₀,z₀)=(1,0,0)

a*(x-1)+b*(y-0)+c*(z-0) =0

for B

a*(3-1)+b*(-1-0)+c*(-3-0) =0

1) 2*a - b - 3*c =0

for C

a*(1-1)+b*(3-0)+c*(-2-0) =0

2) 3*b - 2*c=0 → b= 2/3*c

replacing in 1)

2*a - 2/3*c - 3*c =0

2*a-11/3*c=0 → a=11/6*c

thus

a²+b²+c² = 1

(11/6*c)²+(2/3*c)²+c² = 1

(121/36+4/9+1)*c² = 1

171/36*c²=1 → c= 6/√171

therefore

a=11/6*c = 11/6*6/√171= 11/√171

b=2/3*c= 2/3*6/√171= 4/√171

then the unit normal vector n will be

n=(a,b,c) = (4/√171,11/√171,6/√171)

5 0

4 years ago

A farmer decides to sell 16 pounds of his small 48 pound potato crop. What percent of the crop did he sell? Write a proportion.

Cerrena [4.2K]

Take 100 divided by 48 times 16 and that's your answer

8 0

4 years ago

Read 2 more answers

A rate is a ratio. True or false

KatRina [158]

A rate is a ratio where time is usually involved such as 60 km/ 1 hour.

So, the answer is true.

8 0

3 years ago

Read 2 more answers

What is the total number of different 11-letter arrangements that can be formed using

PolarNik [594]

Answer:

4989600

Step-by-step explanation:

First, find how many letters there are. There is 11 letters.

Next, find which letters are repeated and if so, how many times. The letters O, R, and H are each repeated twice.

Since there are 11 letters, you must do 11 ! (factorial). There are 3 letters that are repeated twice, so you must divide by 2 ! 3 times.

Your equation is:

11!/ 2! * 2! * 2!

It simplifies into 39,916,800/8 which is equal to 4989600

5 0

3 years ago

Multiple Choice

Zanzabum

A) Answer: A. 22.5 cm, 30 cm, and 37.5 cm

Setting the unity measurement as x, the three sides will be 3·x, 4·x, and 5·x. The sum of the three sides will be (3 + 4 + 5)·x = 12·x; this corresponds to the perimeter, therefore:

12·x = 90 cm

We can solve for x:

x = 90 / 12 = 7.5 cm

Hence, the sides will be:

7.5 · 3 = 22.5 cm

7.5 · 4 = 30 cm

7.5 · 5 = 37.5 cm

This situation is represented in option A (which is also the only option whose sides add up to 90).

B) Answer: C. never true

We have:

6 + 8x - 9 = 11x + 14 - 3x

Bring all the terms with x on the left, by subtracting from both sides 11x and -3x, and all the numbers on the right by subtrcting from both sides 6 and -9:

6 + 8x - 9 - 11x - (-3x) - 6 - (-9) = 11x + 14 - 3x - 11x - (-3x) - 6 - (-9)

Cancel out the opposite terms from each side:

8x - 11x + 3x = 14 - 6 + 9

Combine likely terms:

0x = 17

Now you should ask yourself: what number multiplied by zero gives 17 as a result? The answer is none because a number multiplied by zero always equals zero.

Hence, the statement is never true.

C) Answer: $r \leq \frac{1}{5}$