Question

which of the following best represents \\(\vec{a} + \vec{b} = \vec{c}\\)?

Explanation:

To determine which diagram best represents \(\vec{A} + \vec{B} = \vec{C}\), we use the triangle law of vector addition: To add two vectors \(\vec{A}\) and \(\vec{B}\), place the tail of \(\vec{B}\) at the head of \(\vec{A}\). The resultant vector \(\vec{C}\) starts from the tail of \(\vec{A}\) and ends at the head of \(\vec{B}\).

Step 1: Analyze Vector Directions

• \(\vec{A}\) is downward (vertical).
• \(\vec{B}\) is to the right (horizontal).

Step 2: Apply the Triangle Law

For \(\vec{A} + \vec{B}\):

1. Draw \(\vec{A}\) (downward).
2. Draw \(\vec{B}\) starting from the head of \(\vec{A}\) (to the right).
3. The resultant \(\vec{C}\) connects the tail of \(\vec{A}\) to the head of \(\vec{B}\).

Step 3: Evaluate the Diagrams

• First diagram: \(\vec{B}\) starts at the tail of \(\vec{A}\) (incorrect, violates the triangle law).
• Second diagram: \(\vec{B}\) starts at the head of \(\vec{A}\), and \(\vec{C}\) connects the tail of \(\vec{A}\) to the head of \(\vec{B}\) (matches the triangle law).
• Third diagram: \(\vec{C}\) direction is inconsistent with the triangle law (points upward, but \(\vec{A}\) is downward and \(\vec{B}\) is rightward).

Answer:

The second diagram (where \(\vec{A}\) is downward, \(\vec{B}\) is rightward from \(\vec{A}\)’s head, and \(\vec{C}\) connects \(\vec{A}\)’s tail to \(\vec{B}\)’s head).