3. Choose $$ y $$ in order that the vectors $$ \vec{a}=7 \hat{i}+y \hat{j}+\hat{k}, \vec{b}=3 \hat{i}+2 \hat{j}+\hat{k} $$, $$ \vec{c}=5 \hat{i}+3 \hat{j}+\hat{k} $$ are linearly dependent.

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To determine the value of $$ y $$ that makes the vectors $$ \vec{a} = 7\hat{i} + y\hat{j} + \hat{k} $$, $$ \vec{b} = 3\hat{i} + 2\hat{j} + \hat{k} $$, and $$ \vec{c} = 5\hat{i} + 3\hat{j} + \hat{k} $$ linearly dependent, we can use the concept of the determinant of a matrix formed by these vectors.

### Step 1: Form the Matrix
Arrange the vectors as columns in a matrix:

$$
\begin{pmatrix}
7 & 3 & 5 \
y & 2 & 3 \
1 & 1 & 1
\end{pmatrix}
$$

### Step 2: Calculate the Determinant
For the vectors to be linearly dependent, the determinant of this matrix must be zero.

$$
\text{Determinant} = 7 \begin{vmatrix} 2 & 3 \ 1 & 1 \end{vmatrix} - 3 \begin{vmatrix} y & 3 \ 1 & 1 \end{vmatrix} + 5 \begin{vmatrix} y & 2 \ 1 & 1 \end{vmatrix}
$$

Calculate each 2x2 determinant:

$$
\begin{vmatrix} 2 & 3 \ 1 & 1 \end{vmatrix} = (2)(1) - (3)(1) = -1
$$
$$
\begin{vmatrix} y & 3 \ 1 & 1 \end{vmatrix} = y(1) - 3(1) = y - 3
$$
$$
\begin{vmatrix} y & 2 \ 1 & 1 \end{vmatrix} = y(1) - 2(1) = y - 2
$$

Substitute back into the determinant:

$$
\text{Determinant} = 7(-1) - 3(y - 3) + 5(y - 2) = -7 - 3y + 9 + 5y - 10
$$

Combine like terms:

$$
-7 + 9 - 10 + (-3y + 5y) = -8 + 2y
$$

### Step 3: Set the Determinant to Zero

$$
-8 + 2y = 0 \implies 2y = 8 \implies y = 4
$$

**Answer:**  
$$ y = 4 $$
$$ y = 4 $$

Choose in order that the vectors ,
are linearly dependent.

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Choose in order that the vectors , are linearly dependent.