Vector Addition Worksheet demonstrating the component method to find resultant displacement for vectors A (2.3 km at 45 degrees) and B (4.8 km at 90 degrees).

To solve the problem, we need to analyze each resultant vector in the given diagrams and express it in terms of the vectors \(\vec{P}\), \(\vec{Q}\), and \(\vec{R}\). Let's go through each part step by step.

Given Vectors:

- \(\vec{P}\): A horizontal vector pointing to the right.
- \(\vec{Q}\): A diagonal vector pointing upward and to the right.
- \(\vec{R}\): A vertical vector pointing downward.

Task:

Write an expression for each resultant vector using \(\vec{P}\), \(\vec{Q}\), and \(\vec{R}\).

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Part A:

#### Diagram:
- The resultant vector \(\vec{A}\) is formed by adding \(\vec{P}\) and \(\vec{Q}\).
- \(\vec{P}\) is horizontal.
- \(\vec{Q}\) is diagonal, starting from the tip of \(\vec{P}\).

#### Expression:
\[
\vec{A} = \vec{P} + \vec{Q}
\]

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Part B:

#### Diagram:
- The resultant vector \(\vec{B}\) is formed by subtracting \(\vec{Q}\) from \(\vec{P}\).
- \(\vec{P}\) is horizontal.
- \(\vec{Q}\) is diagonal, but it is reversed (pointing downward and to the left) because it is being subtracted.

#### Expression:
\[
\vec{B} = \vec{P} - \vec{Q}
\]

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Part C:

#### Diagram:
- The resultant vector \(\vec{C}\) is formed by adding \(\vec{Q}\) and \(\vec{R}\).
- \(\vec{Q}\) is diagonal.
- \(\vec{R}\) is vertical, starting from the tip of \(\vec{Q}\).

#### Expression:
\[
\vec{C} = \vec{Q} + \vec{R}
\]

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Part D:

#### Diagram:
- The resultant vector \(\vec{D}\) is formed by subtracting \(\vec{R}\) from \(\vec{Q}\).
- \(\vec{Q}\) is diagonal.
- \(\vec{R}\) is vertical, but it is reversed (pointing upward) because it is being subtracted.

#### Expression:
\[
\vec{D} = \vec{Q} - \vec{R}
\]

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Part E:

#### Diagram:
- The resultant vector \(\vec{E}\) is formed by adding \(\vec{P}\) and \(\vec{R}\).
- \(\vec{P}\) is horizontal.
- \(\vec{R}\) is vertical, starting from the tip of \(\vec{P}\).

#### Expression:
\[
\vec{E} = \vec{P} + \vec{R}
\]

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Final Answers:

\[
\boxed{
\begin{aligned}
\vec{A} &= \vec{P} + \vec{Q} \\
\vec{B} &= \vec{P} - \vec{Q} \\
\vec{C} &= \vec{Q} + \vec{R} \\
\vec{D} &= \vec{Q} - \vec{R} \\
\vec{E} &= \vec{P} + \vec{R}
\end{aligned}
}
\]