The problem asks us to show that the points $A(-2i + 3j + 5k)$, $B(i + 2j + 3k)$, and $C(7i - k)$ are collinear. Collinear points lie on the same line.

1. Problem Description

The problem asks us to show that the points $A(-2i + 3j + 5k)$, $B(i + 2j + 3k)$, and $C(7i - k)$ are collinear. Collinear points lie on the same line.

2. Solution Steps

To determine if the points $A, B, C$ are collinear, we can check if the vectors $\vec{AB}$ and $\vec{AC}$ are parallel. Two vectors are parallel if one is a scalar multiple of the other.

First, we find the vector $\vec{AB}$:

$\vec{AB} = B - A = (i + 2j + 3k) - (-2i + 3j + 5k) = (1 - (-2))i + (2 - 3)j + (3 - 5)k = 3i - j - 2k$

Next, we find the vector $\vec{AC}$:

$\vec{AC} = C - A = (7i - k) - (-2i + 3j + 5k) = (7 - (-2))i + (0 - 3)j + (-1 - 5)k = 9i - 3j - 6k$

Now we check if $\vec{AC}$ is a scalar multiple of $\vec{AB}$.

Let's see if there exists a scalar $c$ such that $\vec{AC} = c\vec{AB}$.

$9i - 3j - 6k = c(3i - j - 2k) = 3ci - cj - 2ck$

Equating the coefficients of $i, j, k$:

$9 = 3c$

$-3 = -c$

$-6 = -2c$

From each of these equations, we find $c = 3$.

Since there exists a scalar $c = 3$ such that $\vec{AC} = 3\vec{AB}$, the vectors $\vec{AB}$ and $\vec{AC}$ are parallel. Since they share the point $A$, the points $A, B, C$ are collinear.

3. Final Answer

The points A, B, and C are collinear.