If three points A,B and C have position vectors (1,x,3),(3,4,7) and (y,-2,-5), respectively and if they are collinear, then find (x,y).

Answer

To solve the problem of finding the values of \(x\) and \(y\) such that the points \(A\), \(B\), and \(C\) with position vectors \((1, x, 3)\), \((3, 4, 7)\), and \((y, -2, -5)\) respectively are collinear, we can follow these steps:

### Step 1: Define the vectors
We first define the vectors \( \vec{AB} \) and \( \vec{AC} \).

\[ \vec{AB} = \vec{B} - \vec{A} = (3 - 1, 4 - x, 7 - 3) = (2, 4 - x, 4) \]

\[ \vec{AC} = \vec{C} - \vec{A} = (y - 1, -2 - x, -5 - 3) = (y - 1, -2 - x, -8) \]

### Step 2: Set up the collinearity condition
Since points \(A\), \(B\), and \(C\) are collinear, we can express this as:

\[ \vec{AB} = \lambda \vec{AC} \]

This gives us the following equations by equating the components:

1. \(2 = \lambda (y - 1)\) (i-component)
2. \(4 - x = \lambda (-2 - x)\) (j-component)
3. \(4 = \lambda (-8)\) (k-component)

### Step 3: Solve for \(\lambda\) from the k-component
From the third equation:

\[ \lambda = \frac{4}{-8} = -\frac{1}{2} \]

### Step 4: Substitute \(\lambda\) into the other equations
Now, substitute \(\lambda = -\frac{1}{2}\) into the first equation:

\[ 2 = -\frac{1}{2}(y - 1) \]

Multiplying both sides by -2 gives:

\[ -4 = y - 1 \implies y = -3 \]

### Step 5: Substitute \(\lambda\) into the second equation
Now substitute \(\lambda = -\frac{1}{2}\) into the second equation:

\[ 4 - x = -\frac{1}{2}(-2 - x) \]

This simplifies to:

\[ 4 - x = \frac{1}{2}(2 + x) \]

Multiplying both sides by 2 to eliminate the fraction:

\[ 8 - 2x = 2 + x \]

Rearranging gives:

\[ 8 - 2 = 2x + x \implies 6 = 3x \implies x = 2 \]

### Final Result
Thus, the values of \(x\) and \(y\) are:

\[ (x, y) = (2, -3) \]