different questions

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July 2nd 2007, 02:40 PM
Raiden_11

different questions

I need help with these questions.....they're from grade 12 geometry and discrete math.

1. Analyse the normal vectors of the following planes and describe the situation of each system. Include a geometric interpretation of each solution:

7x-5y+4z-9=0

-5x+4y+z-3=0

x+2y-z+1=0

2. Analyse the normal vectors of the following planes and describe the situation of each system. Include a geometric interpretation of each solution:

7x-5y+4z-9=0

-x+3y+5z-2=0

5x+y+14z-20=0
July 2nd 2007, 09:44 PM
earboth

1 Attachment(s)

Quote:

Originally Posted by Raiden_11

I need help with these questions.....they're from grade 12 geometry and discrete math.

1. Analyse the normal vectors of the following planes and describe the situation of each system. Include a geometric interpretation of each solution:

$p_1:7x-5y+4z-9=0$
$p_2: -5x+4y+z-3=0$
$p_3: x+2y-z+1=0$

...

Hello,

the normal vectors of the planes are: $\overrightarrow{n_1}=\left(\begin{array}{c}7\\-5\\4\end{array} \right)$ , $\overrightarrow{n_2}=\left(\begin{array}{c}-5\\4\\1\end{array} \right)$ , $\overrightarrow{n_3}=\left(\begin{array}{c}1\\2\\-1\end{array} \right)$
These vectors are linear independent(?) and the planes have one common point $P\left( \frac{4}{13}, \frac{7}{13}, \frac{31}{13} \right)$

Quote:

Originally Posted by Raiden_11

I need help with these questions.....they're from grade 12 geometry and discrete math.
...
2. Analyse the normal vectors of the following planes and describe the situation of each system. Include a geometric interpretation of each solution:

7x-5y+4z-9=0
-x+3y+5z-2=0
5x+y+14z-20=0

the normal vectors of the planes are: $\overrightarrow{n_1}=\left(\begin{array}{c}7\\-5\\4\end{array} \right)$ , $\overrightarrow{n_2}=\left(\begin{array}{c}-1\\3\\5\end{array} \right)$ , $\overrightarrow{n_3}=\left(\begin{array}{c}5\\1\\1 4\end{array} \right)$
These vectors are linear dependent(?) because the following equation has a unique solution:

$r \cdot \left(\begin{array}{c}7\\-5\\4\end{array} \right) + s \cdot \left(\begin{array}{c}-1\\3\\5\end{array} \right) = \left(\begin{array}{c}5\\1\\14\end{array} \right)$ . This equation is true for r = 1 and s = 2.

First guess: The planes intersect in a common line.

According to my calculations the three planes are not parallel and there doesn't exist a common line thus they intersect in three different parallel lines.

EDIT: I've attached a diagram of the three planes.
July 3rd 2007, 12:30 PM
Raiden_11

1 Attachment(s)

Can u help me with these?

1. On page 345 of the textbook, the diagonal pattern of Pascal’s Triangle illustrates that C(7, 5) = C(4, 4) + C(5, 4) + C(6, 4). Prove this relationship using the following methods.

a)numerically by using factorials
b)by reasoning, using the meaning of combinations

Attachment 3479
July 3rd 2007, 12:31 PM
Raiden_11

1 Attachment(s)

heres another one

2. Observe the relationship between the elements of Diagonal 1 and Diagonal 2 in Pascal’s Triangle, shown on the next page. The sum of the first two elements in Diagonal 1 is equal to the second element in Diagonal 2. Similarly, the sum of the first 5 elements of Diagonal 1 is equal to the 5th element of Diagonal 2. Suppose we wish to use Pascal’s Triangle to find the sum of the first n natural numbers. Use combinatorial notation to express the sum of the n elements of Diagonal 1 in terms of the corresponding value in Diagonal 2.

Attachment 3480
July 3rd 2007, 12:33 PM
Raiden_11

1 Attachment(s)

heres the last one

3. Using the method of mathematical induction, prove that the following statements are false.
Attachment 3481
July 3rd 2007, 12:35 PM
janvdl

I cant see those last 2 pics.
July 3rd 2007, 09:01 PM
Raiden_11

Is this for #1

Is this for #1 a) and b)
July 3rd 2007, 09:08 PM
Raiden_11

1 Attachment(s)

Can u see #2 now?

2. Observe the relationship between the elements of Diagonal 1 and Diagonal 2 in Pascal’s Triangle, shown on the next page. The sum of the first two elements in Diagonal 1 is equal to the second element in Diagonal 2. Similarly, the sum of the first 5 elements of Diagonal 1 is equal to the 5th element of Diagonal 2. Suppose we wish to use Pascal’s Triangle to find the sum of the first n natural numbers. Use combinatorial notation to express the sum of the n elements of Diagonal 1 in terms of the corresponding value in Diagonal 2.

Attachment 3488
July 3rd 2007, 09:13 PM
Raiden_11

1 Attachment(s)

Can u see #3 now?

3. Using the method of mathematical induction, prove that the following statements are false.

Attachment 3489
July 4th 2007, 02:07 AM
galactus

Quote:

Originally Posted by Raiden_11

Is this for #1 a) and b)

Yes, but I couldn't see it very well and misread it. Done wrong problem.

$\sum_{i=1}^{n}2i=2n^{2}-2n+2$

$2+4+6+8+....+2n=2n^{2}-2n+2$

Show $P_{1}$ is true: 2(1)=2...true

Assume $P_{k}$ is true: $2+4+6+8+....+2k=2k^{2}-2k+2$

Show $P_{k+1}$ is true:

$2+4+6+8+....+2k+2(k+1)=2k^{2}-2k+2+2(k+1)=2k^{2}+4=2(k^{2}+2)$

This shows $P_{k}$ is not true and, therefore, $P_{n}$ is not true.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~

Actually, another case may be to prove what the sum of 2i equals and, therefore, show that the statement is false.

$\sum_{i=1}^{n}2i=n(n+1)$

I will skip ahead of the first two steps and go right to $P_{k+1}$

$2+4+6+8+....+2k+2(k+1)=k(k+1)+2(k+1)=(k+1)(k+2)$

This shows that $P_{k+1}$ is true and, therefore, $P+{n}$ is true and the above statement is false.
July 4th 2007, 10:39 AM
Raiden_11

Which one is a) and Which one is b)?

Which one is a) and which one is b), also can u help me with #2 and #3, i made them bigger for you to see.
July 4th 2007, 11:29 AM
galactus

That was 3a. The first induction you posted.
July 4th 2007, 11:33 AM
Raiden_11

And..what???

oh that was 3a............what about 1 a) and b).....what about 3b) oh man i'm confused!!!!!