Question

In the cube of side $'a'$ shown in the figure, the vector from the central point of the face $ABOD$ to the central point of the face $BEFO$ will be:

• A. $\frac{1}{2} a \left(\hat{i} -\hat{k}\right) $
• B. $\frac{1}{2} a \left(\hat{j} -\hat{i}\right) $
• C. $\frac{1}{2} a \left(\hat{k} -\hat{i}\right) $
• D. $\frac{1}{2} a \left(\hat{j} -\hat{k}\right) $

Answer 1

The Correct Option is B

To solve this problem, we need to determine the vector from the center of the face \( ABOD \) to the center of the face \( BEFO \) of a cube with side length \( a \). Let's examine each step carefully:

Step 1: Identify the Centers of the Faces

1. The face \( ABOD \) is in the plane containing points \( A, B, O, \) and \( D \). Since it's a face of a cube, it's a square with side length \( a \). The center of \( ABOD \) would be the average of the coordinates of these points.

Assuming the cube is aligned with the coordinate axes, we can place it such that:

• \( A = (0, 0, 0) \)
• \( B = (a, 0, 0) \)
• \( O = (a, 0, a) \)
• \( D = (0, 0, a) \)

The center of the face \( ABOD \), therefore, is given by:

\(\text{Center of } ABOD = \left(\frac{0 + a + a + 0}{4}, \frac{0 + 0 + 0 + 0}{4}, \frac{0 + 0 + a + a}{4}\right) = \left(\frac{a}{2}, 0, \frac{a}{2}\right)\)

Step 2: Compute the Center of the Face \( BEFO \)

1. The face \( BEFO \) is a square in the plane containing points \( B, E, F, \) and \( O \).

Assuming the cube is aligned with the coordinate axes, we can place it such that:

• \( E = (a, a, 0) \)
• \( F = (a, a, a) \)

The center of the face \( BEFO \) is given by:

\(\text{Center of } BEFO = \left(\frac{a + a + a + a}{4}, \frac{0 + a + a + 0}{4}, \frac{0 + 0 + a + a}{4}\right) = \left(a, \frac{a}{2}, \frac{a}{2}\right)\)

Step 3: Calculate the Vector from \( ABOD \) to \( BEFO \)

1. To find the vector, subtract the central coordinates of \( ABOD \) from \( BEFO \):
2. The required vector is:

(\text{Vector}) = \left(a - \frac{a}{2}, \frac{a}{2} - 0, \frac{a}{2} - \frac{a}{2}\right) = \left(\frac{a}{2}, \frac{a}{2}, 0\right)

This vector can be represented in terms of unit vectors \( \hat{i}, \hat{j}, \text{ and } \hat{k} \) as:

=\frac{1}{2}a \hat{j} - \frac{1}{2}a \hat{i} \text{ (since } \hat{k \text{ component is zero)}}\)

Thus, the correct answer is:

\(\frac{1}{2} a (\hat{j} - \hat{i})\)