To solve this problem, we need to find the component form of the vector [tex]$\overrightarrow{PQ} + 3 \overrightarrow{RS}$[/tex].
### Step 1: Calculate the vector [tex]$\overrightarrow{PQ}$[/tex]
Given points [tex]\( P = (5, 4) \)[/tex] and [tex]\( Q = (7, 3) \)[/tex], the vector [tex]\(\overrightarrow{PQ}\)[/tex] is calculated as:
[tex]\[
\overrightarrow{PQ} = (Q_x - P_x, Q_y - P_y)
\][/tex]
Substituting the coordinates of [tex]\(P\)[/tex] and [tex]\(Q\)[/tex]:
[tex]\[
\overrightarrow{PQ} = (7 - 5, 3 - 4) = (2, -1)
\][/tex]
### Step 2: Calculate the vector [tex]$\overrightarrow{RS}$[/tex]
Given points [tex]\( R = (8, 6) \)[/tex] and [tex]\( S = (4, 1) \)[/tex], the vector [tex]\(\overrightarrow{RS}\)[/tex] is calculated as:
[tex]\[
\overrightarrow{RS} = (S_x - R_x, S_y - R_y)
\][/tex]
Substituting the coordinates of [tex]\(R\)[/tex] and [tex]\(S\)[/tex]:
[tex]\[
\overrightarrow{RS} = (4 - 8, 1 - 6) = (-4, -5)
\][/tex]
### Step 3: Calculate 3 times the vector [tex]$\overrightarrow{RS}$[/tex]
To find [tex]\(3 \overrightarrow{RS}\)[/tex], we multiply each component of [tex]\(\overrightarrow{RS}\)[/tex] by 3:
[tex]\[
3 \overrightarrow{RS} = 3 \cdot (-4, -5) = (3 \cdot -4, 3 \cdot -5) = (-12, -15)
\][/tex]
### Step 4: Calculate the resultant vector [tex]$\overrightarrow{PQ} + 3 \overrightarrow{RS}$[/tex]
Now we add the components of [tex]\(\overrightarrow{PQ}\)[/tex] and [tex]\(3 \overrightarrow{RS}\)[/tex]:
[tex]\[
\overrightarrow{PQ} + 3 \overrightarrow{RS} = (2, -1) + (-12, -15)
\][/tex]
Adding the corresponding components:
[tex]\[
(2 + (-12), -1 + (-15)) = (-10, -16)
\][/tex]
Thus, the component form of the vector [tex]\(\overrightarrow{PQ} + 3 \overrightarrow{RS}\)[/tex] is [tex]\(\langle -10, -16 \rangle\)[/tex].
Among the given choices, the correct answer is:
[tex]\[
\boxed{\langle -10, -16 \rangle}
\][/tex]
