Question

Given that upper P equals left parenthesis 5 comma 4 right parenthesis, upper Q equals left parenthesis 7 comma 3 right parenthesis, upper R equals left parenthesis 8 comma 6 right parenthesis, and upper S equals left parenthesis 4 comma 1 right parenthesis

Use the information from part A to find the magnitude of the vector Modifying above upper P upper Q with right arrow plus 3 Modifying above upper R upper S with right arrow.
(1 point)

Answer 1

To find the magnitude of the vector \( \vec{PQ} + 3\vec{RS} \), we first need to calculate each vector.

**Step 1: Calculate \( \vec{PQ} \)**

The vector \( \vec{PQ} \) is calculated by subtracting the coordinates of point \( P \) from point \( Q \):

\[
\vec{PQ} = Q - P = (7, 3) - (5, 4) = (7 - 5, 3 - 4) = (2, -1)
\]

**Step 2: Calculate \( \vec{RS} \)**

Next, we calculate the vector \( \vec{RS} \) as follows:

\[
\vec{RS} = S - R = (4, 1) - (8, 6) = (4 - 8, 1 - 6) = (-4, -5)
\]

**Step 3: Calculate \( 3\vec{RS} \)**

Now, we multiply the vector \( \vec{RS} \) by 3:

\[
3\vec{RS} = 3 \cdot (-4, -5) = (-12, -15)
\]

**Step 4: Add the two vectors**

Now we will add \( \vec{PQ} \) and \( 3 \vec{RS} \):

\[
\vec{PQ} + 3\vec{RS} = (2, -1) + (-12, -15) = (2 - 12, -1 - 15) = (-10, -16)
\]

**Step 5: Calculate the magnitude of the resulting vector**

The magnitude \( |\vec{v}| \) of a vector \( \vec{v} = (x, y) \) is calculated as:

\[
|\vec{v}| = \sqrt{x^2 + y^2}
\]

For our resulting vector \( (-10, -16) \):

\[
|\vec{PQ} + 3\vec{RS}| = \sqrt{(-10)^2 + (-16)^2} = \sqrt{100 + 256} = \sqrt{356}
\]

To simplify \( \sqrt{356} \):

\[
\sqrt{356} = \sqrt{4 \cdot 89} = 2\sqrt{89}
\]

Thus, the magnitude of the vector \( \vec{PQ} + 3\vec{RS} \) is \( 2\sqrt{89} \).

So the final answer is:

\[
\boxed{2\sqrt{89}}
\]