Consider the following statements with respect to a vector d = (a × b) × c:

I. d is coplanar with a and b.

II. d is perpendicular to c.

Which of the statements given above is/are correct?

Calculation:

Given,

The vector \( \vec{d} = (\vec{a} \times \vec{b}) \times \vec{c} \)

Statement I: \( \vec{d} \) is coplanar with \( \vec{a} \) and \( \vec{b} \).

We use the vector triple product identity: \( (\vec{a} \times \vec{b}) \times \vec{c} = (\vec{a} \cdot \vec{c}) \vec{b} - (\vec{b} \cdot \vec{c}) \vec{a} \).

This shows that \( \vec{d} \) is a linear combination of \( \vec{a} \) and \( \vec{b} \), hence \( \vec{d} \) is coplanar with \( \vec{a} \) and \( \vec{b} \).

Therefore, Statement I is correct.

Statement II: \( \vec{d} \) is perpendicular to \( \vec{c} \).

To check this, compute the dot product \( \vec{d} \cdot \vec{c} \). Using the vector triple product identity, we find:

\( \vec{d} \cdot \vec{c} = (\vec{a} \cdot \vec{c})(\vec{b} \cdot \vec{c}) - (\vec{b} \cdot \vec{c})(\vec{a} \cdot \vec{c}) = 0 \),

which means \( \vec{d} \) is perpendicular to \( \vec{c} \).

Therefore, Statement II is correct.

∴ Both Statement I and Statement II are correct.

Hence, the correct answer is option 3.