) \cdot \vec{c}$$

Here, $\vec{a} \times \vec{b}$ gives a vector.

Then dot product of that vector with $\vec{c}$ is possible.

So A is meaningful.

For expression B,

$$\vec{a} \times (\vec{b} \cdot \vec{c})$$

Here, $\vec{b} \cdot \vec{c}$ gives a scalar.

But cross product is defined between two vectors, not between a vector and a scalar.

So B is meaningless.

For expression C,

$$\vec{a} \cdot (\vec{b} \cdot \vec{c})$$

Again, $\vec{b} \cdot \vec{c}$ is a scalar.

Dot product is defined between two vectors, not between a vector and a scalar.

So C is meaningless.

For expression D,

$$|\vec{a}|(\vec{b} \cdot \vec{c})$$

Here, $|\vec{a}|$ is a scalar, and $\vec{b} \cdot \vec{c}$ is also a scalar.

Multiplication of two scalars is meaningful.

So D is meaningful.

For expression E,

$$(\vec{a} \cdot \vec{b}) \times (\vec{b} \cdot \vec{c})$$

Both $\vec{a} \cdot \vec{b}$ and $\vec{b} \cdot \vec{c}$ are scalars.

Cross product is not defined between scalars.

So E is meaningless.

Thus:

A and D are meaningful.

B, C and E are meaningless.

So the statement “B and E are meaningful” is not correct.

Hence, the correct answer is:

$$\boxed{\text{Option A}}$$

Vector Algebra

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