Question Details

$\overset{}{O A} BC is a parallelogram. If \vec{O B} = \vec{a} and \vec{A C} = \vec{b}, then \vec{O A} is equal to$

Options

$\vec{a} - \vec{b}$

$\vec{(a} - \vec{b}$ )/2

$\vec{a} \vec{+ b}$

$\vec{(a} + \vec{b}$ )/2

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Correct Answer :

Option B

$\vec{(a} - \vec{b}$ )/2

Solution :

The correct answer is:
$\frac{(\vec{a} - \vec{b})}{2}$

1. Image Analysis and Given Information:
By analyzing the provided image, we can identify a parallelogram labeled counter-clockwise with vertices O (at the bottom-left), A (at the bottom-right), B (at the top-right), and C (at the top-left).
The diagonals of the parallelogram are shown with directional arrows indicating vector paths:
- The diagonal from O to B is labeled with an arrow pointing towards B, representing the vector:
$\vec{O B} = \vec{a}$
- The diagonal from A to C is labeled with an arrow pointing towards C, representing the vector:
$\vec{A C} = \vec{b}$

2. Defining the Vector Relations:
Let us define the sides of the parallelogram as vectors. Let O be the origin.
Let:
$\vec{O A} = \vec{u}$
and
$\vec{O C} = \vec{v}$
Since OABC is a parallelogram, the opposite sides are equal and parallel. Therefore:
$\vec{C B} = \vec{O A} = \vec{u}$
and
$\vec{A B} = \vec{O C} = \vec{v}$

3. Setting up the Vector Equations:
Using the triangle law of vector addition on triangle OAB:
$\vec{O B} = \vec{O A} + \vec{A B}$
Substituting the defined vectors:
$\vec{a} = \vec{u} + \vec{v}$ (Equation 1)
Similarly, applying vector subtraction to express the diagonal vector $\vec{A C}$ in terms of the position vectors of C and A:
$\vec{A C} = \vec{O C} - \vec{O A}$
Substituting the defined vectors:
$\vec{b} = \vec{v} - \vec{u}$ (Equation 2)

4. Solving for Vector OA:
We want to find vector $\vec{O A}$ , which is defined as $\vec{u}$ .
To isolate $\vec{u}$ , we subtract Equation 2 from Equation 1:
$\vec{a} - \vec{b} = (\vec{u} + \vec{v}) - (\vec{v} - \vec{u})$
Simplifying the right-hand side:
$\vec{a} - \vec{b} = \vec{u} + \vec{v} - \vec{v} + \vec{u}$
$\vec{a} - \vec{b} = 2 \vec{u}$
Dividing both sides by 2:
$\vec{u} = \frac{\vec{a} - \vec{b}}{2}$
Thus, the vector $\vec{O A}$ is equal to:
$\frac{(\vec{a} - \vec{b})}{2}$