Question

Point A, B, C position vectors $3i-yj+2k$, $5i-j+k$ and $3xi+3j-k$ are collinear. The values of $x$ and $y$ came out to be as $x=3/8$ and $y=5$. Choose the correct option.

Answer 1

The values of x and y given in the question are incorrect.

Answer 2

To determine if the given values of $x$ and $y$ are correct for collinear points, we use the property that if three points A, B, and C are collinear, then the vector $\vec{AB}$ must be parallel to the vector $\vec{BC}$. This implies that $\vec{AB} = k \vec{BC}$ for some scalar $k$.

Let the position vectors of points A, B, and C be $\vec{a}$, $\vec{b}$, and $\vec{c}$ respectively.

Given: $\vec{a} = 3\mathbf{i} - y\mathbf{j} + 2\mathbf{k}$

$\vec{b} = 5\mathbf{i} - \mathbf{j} + \mathbf{k}$

$\vec{c} = 3x\mathbf{i} + 3\mathbf{j} - \mathbf{k}$

First, calculate the vectors $\vec{AB}$ and $\vec{BC}$:

$\vec{AB} = \vec{b} - \vec{a} = (5-3)\mathbf{i} + (-1 - (-y))\mathbf{j} + (1-2)\mathbf{k}$ $\vec{AB} = 2\mathbf{i} + (y-1)\mathbf{j} - \mathbf{k}$

$\vec{BC} = \vec{c} - \vec{b} = (3x-5)\mathbf{i} + (3 - (-1))\mathbf{j} + (-1-1)\mathbf{k}$ $\vec{BC} = (3x-5)\mathbf{i} + 4\mathbf{j} - 2\mathbf{k}$

For points A, B, C to be collinear, $\vec{AB}$ must be a scalar multiple of $\vec{BC}$. So, $\vec{AB} = k \vec{BC}$ $2\mathbf{i} + (y-1)\mathbf{j} - \mathbf{k} = k((3x-5)\mathbf{i} + 4\mathbf{j} - 2\mathbf{k})$

Equating the corresponding components:

1. For the $\mathbf{k}$ component: $-1 = k(-2)$ $k = \frac{-1}{-2} = \frac{1}{2}$

2. For the $\mathbf{i}$ component: $2 = k(3x-5)$ Substitute $k = \frac{1}{2}$: $2 = \frac{1}{2}(3x-5)$ $4 = 3x-5$ $3x = 9$ $x = 3$

3. For the $\mathbf{j}$ component: $y-1 = k(4)$ Substitute $k = \frac{1}{2}$: $y-1 = \frac{1}{2}(4)$ $y-1 = 2$ $y = 3$

Thus, for the points A, B, C to be collinear, the values of $x$ and $y$ must be $x=3$ and $y=3$.

The question states that the values of $x$ and $y$ came out to be $x=3/8$ and $y=5$.

Comparing our calculated values ($x=3, y=3$) with the given values ($x=3/8, y=5$), we find that they do not match. Therefore, the statement given in the question is incorrect.

In summary: The values of $x$ and $y$ that make points A, B, C collinear are $x=3$ and $y=3$. The values provided in the question ($x=3/8$ and $y=5$) are incorrect.