Use cross products to determine whether the points $$ A, B $$, and $$ C $$ are collinear. $$ A(-7,-7,1), B(1,4,7) $$, and $$ C(5,10,15) $$ Are the points $$ A, B $$, and $$ C $$ collinear? A. No, because the cross product of the vectors $$ \overrightarrow{A B} $$ and $$ \overrightarrow{A C} $$, is not 0 . B. No, because the cross product of the vectors $$ \overrightarrow{A B} $$ and $$ \overrightarrow{A C} $$, is 0 . C. Yes, because the cross product of the vectors $$ \overrightarrow{A B} $$ and $$ \overrightarrow{A C} $$, is 0 . D. Yes, because the cross product of the vectors $$ \overrightarrow{A B} $$ and $$ \overrightarrow{A C} $$, is not 0 .

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To determine whether the points $$ A(-7,-7,1) $$, $$ B(1,4,7) $$, and $$ C(5,10,15) $$ are collinear, we'll use the cross product of vectors $$ \overrightarrow{AB} $$ and $$ \overrightarrow{AC} $$.

**Step 1: Calculate the vectors $$ \overrightarrow{AB} $$ and $$ \overrightarrow{AC} $$.**

$$
\overrightarrow{AB} = B - A = (1 - (-7),\, 4 - (-7),\, 7 - 1) = (8,\, 11,\, 6)
$$

$$
\overrightarrow{AC} = C - A = (5 - (-7),\, 10 - (-7),\, 15 - 1) = (12,\, 17,\, 14)
$$

**Step 2: Compute the cross product $$ \overrightarrow{AB} \times \overrightarrow{AC} $$.**

$$
\overrightarrow{AB} \times \overrightarrow{AC} = 
\begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \
8 & 11 & 6 \
12 & 17 & 14 \
\end{vmatrix}
$$

$$
= \mathbf{i}(11 \times 14 - 6 \times 17) - \mathbf{j}(8 \times 14 - 6 \times 12) + \mathbf{k}(8 \times 17 - 11 \times 12)
$$

$$
= \mathbf{i}(154 - 102) - \mathbf{j}(112 - 72) + \mathbf{k}(136 - 132)
$$

$$
= \mathbf{i}(52) - \mathbf{j}(40) + \mathbf{k}(4)
$$

$$
= (52, -40, 4)
$$

**Step 3: Analyze the cross product.**

The cross product $$ \overrightarrow{AB} \times \overrightarrow{AC} = (52, -40, 4) $$ is **not** the zero vector. This indicates that vectors $$ \overrightarrow{AB} $$ and $$ \overrightarrow{AC} $$ are not parallel, and therefore, the points $$ A $$, $$ B $$, and $$ C $$ are **not** collinear.

**Conclusion:**

The correct answer is:

**A. No, because the cross product of the vectors $$ \overrightarrow{AB} $$ and $$ \overrightarrow{AC} $$, is not 0.**

**Answer:**  
A. No, because the cross product of the vectors $$ \overrightarrow{A B} $$ and $$ \overrightarrow{A C} $$, is not 0 .
The points $$ A $$, $$ B $$, and $$ C $$ are **not** collinear because the cross product of vectors $$ \overrightarrow{AB} $$ and $$ \overrightarrow{AC} $$ is not zero.

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