Question

Find the value of \( \hat{i} \cdot (\hat{j} \times \hat{k}) + \hat{j} \cdot (\hat{i} \times \hat{k}) + \hat{k} \cdot (\hat{i} \times \hat{j}) \).

Hint:

The scalar triple product \( [\hat{i} \ \hat{j} \ \hat{k}] = \hat{i} \cdot (\hat{j} \times \hat{k}) \) is equal to the determinant of the matrix formed by these vectors, which is the identity matrix, so its determinant is 1. Swapping any two vectors negates the result. In the second term, \( \hat{i} \) and \( \hat{j} \) are swapped relative to the cyclic order, hence the -1.

Answer 1

Step 1: Understanding the Concept:
This problem involves the scalar triple product and the properties of the standard orthogonal unit vectors \( \hat{i}, \hat{j}, \hat{k} \).
The expression \( \vec{a} \cdot (\vec{b} \times \vec{c}) \) is known as the scalar triple product.
Step 2: Key Formula or Approach:
We need to use the cyclic properties of the cross product of unit vectors:
\( \hat{i} \times \hat{j} = \hat{k} \)
\( \hat{j} \times \hat{k} = \hat{i} \)
\( \hat{k} \times \hat{i} = \hat{j} \)
And the anti-cyclic properties:
\( \hat{j} \times \hat{i} = -\hat{k} \)
\( \hat{k} \times \hat{j} = -\hat{i} \)
\( \hat{i} \times \hat{k} = -\hat{j} \)
We also use the dot product properties: \( \hat{i} \cdot \hat{i} = \hat{j} \cdot \hat{j} = \hat{k} \cdot \hat{k} = 1 \).
Step 3: Detailed Explanation:
Let's evaluate each term of the expression separately.
Term 1: \( \hat{i} \cdot (\hat{j} \times \hat{k}) \) Using the cyclic property, \( \hat{j} \times \hat{k} = \hat{i} \). \[ \hat{i} \cdot (\hat{i}) = 1 \] Term 2: \( \hat{j} \cdot (\hat{i} \times \hat{k}) \) Using the anti-cyclic property, \( \hat{i} \times \hat{k} = -\hat{j} \). \[ \hat{j} \cdot (-\hat{j}) = -(\hat{j} \cdot \hat{j}) = -1 \] Term 3: \( \hat{k} \cdot (\hat{i} \times \hat{j}) \) Using the cyclic property, \( \hat{i} \times \hat{j} = \hat{k} \). \[ \hat{k} \cdot (\hat{k}) = 1 \] Now, add the values of the three terms: \[ \text{Value} = 1 + (-1) + 1 = 1 \] Step 4: Final Answer:
The value of the given expression is 1.