I'm stuck on a question in first-year University Maths. I'll put down what I know where I've done it.

Question: Consider these linear equations:

x + 2y - z = 2

2x + y + 3z = 5

x + 5y - 6z = 1

for the planes P1, P2 and P3 respectively.

(a) Using Gaussian elimination, show that there is one free variable, and that the solutions lie along a line L in three dimensions.

Obviously, I'm not going to put down the whole algorithm, but my solution set is as follows:

(S1,S2,S3) = ( (8-7r)/3 , (5r-1)/3 , r )

Where r is our free variable.

Any ideas on how to show (prove) that the equation lies along a line in three dimensions? Or will that be enough?

(b) Write the equation for L in vector form, and find a vector

**b**parallel to the line L.

Would the equation for L in vector form just be [(8-7r)/3]

**i**+ [(5r-1)/3]

**j**+ r

**k**?

Vector

**b**is parallel to that if you just add a constant, correct?

(c) (The tricky one).

Write down normal vectors

**n1**,

**n2**,

**n3**for the planes P1, P2, P3 respectively. Obtain a vector

**c**that lies along the line of intersection of P1 and P2, by using the vectors

**n1**and

**n2**.

Is

**c**parallel to the vector

**b**in (b) above? Should it be?

I have got

**n1**,

**n2**and

**n3**through finding the vector equation for P1, P2 and P3, and using cross-multiplication.

The vector forms are:

P1 - 2

**i**+

**j**- 2

**k**

P2 - (2/5)

**i**+ (1/5)

**j**+ (3/5)

**k**

P3 -

**i**+ (1/5)

**j**- (1/6)

**k**

Using X, Y and Z intercepts, and labelling each point A, B, and C respectively, I obtained the following vectors (I'll only show for normal vector for P1, too much to type here!)

**A->B**= (-2, 1, 0)

**A->C**= (-2, 0, -2)

Cross multiply these and you'll get:

**n1**= 2

**i**+ 4

**j**- 2

**k**

Also,

n2 = (3/25)

**i**+ (6/25)

**j**+ (2/25)

**k**

n3 = (-1/30)

**i**- (1/6)

**j**+ (1/5)

**k**

Is anyone able to just check my work (to make sure the response to my problem is correct), and able to explain to me clearly about how to obtain the vector

**c**and if it should be parallel to vector

**b**as in Part B.

All help is GREATLY appreciated!!!