Question

Find the value of the scalar triple product: \[ \hat{i} \cdot (\hat{j} \times \hat{k}) \]

• A. 1
• B. 0
• C. -1
• D. \( \hat{i} \)

Hint:

The scalar triple product \( \vec{a} \cdot (\vec{b} \times \vec{c}) \) gives the signed volume of the parallelepiped formed by the vectors. For unit vectors \( \hat{i}, \hat{j}, \hat{k} \), their scalar triple product is always 1.

Answer 1

The Correct Option is A

The scalar triple product of three vectors \( \vec{a}, \vec{b}, \vec{c} \) is given by: \[ \vec{a} \cdot (\vec{b} \times \vec{c}) \] It represents the volume of the parallelepiped formed by the vectors \( \vec{a}, \vec{b}, \vec{c} \). For the unit vectors \( \hat{i}, \hat{j}, \hat{k} \), the cross product \( \hat{j} \times \hat{k} \) results in \( \hat{i} \) (from the right-hand rule and the cyclic property of unit vectors): \[ \hat{j} \times \hat{k} = \hat{i} \] Now, taking the dot product of \( \hat{i} \) with \( \hat{i} \): \[ \hat{i} \cdot \hat{i} = 1 \] Thus, the value of the scalar triple product is: \[ \boxed{1} \]