Let ABCD is a parallelogram. If AB= `vec(a)` and BC= `vec(b)` then what is BD equal to ?

Options

A. a) vec(a) + vec(b)
B. b) vec(a) - vec(b)
C. c) -vec(a) - vec(b)
D. d) -vec(a) + vec(b)

Answer

To find the vector \( \vec{BD} \) in the parallelogram ABCD, we can follow these steps:

### Step 1: Understand the Parallelogram Properties
In a parallelogram, opposite sides are equal and parallel. Therefore, we have:
- \( \vec{AB} = \vec{a} \)
- \( \vec{BC} = \vec{b} \)
- \( \vec{AD} = \vec{b} \) (since \( \vec{AD} \) is equal to \( \vec{BC} \))
- \( \vec{CD} = \vec{a} \) (since \( \vec{CD} \) is equal to \( \vec{AB} \))

### Step 2: Express the Vector \( \vec{BD} \)
To find \( \vec{BD} \), we can use the relationship between the vectors:
\[ \vec{BD} = \vec{BA} + \vec{AD} \]

### Step 3: Substitute the Known Vectors
We know that:
- \( \vec{BA} = -\vec{AB} = -\vec{a} \)
- \( \vec{AD} = \vec{b} \)

Substituting these into the equation for \( \vec{BD} \):
\[ \vec{BD} = -\vec{a} + \vec{b} \]

### Step 4: Final Expression
Thus, we can express \( \vec{BD} \) as:
\[ \vec{BD} = \vec{b} - \vec{a} \]

### Conclusion
The vector \( \vec{BD} \) is equal to \( \vec{b} - \vec{a} \).

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