<p>If \(\vec{a} = \dfrac{1}{\sqrt{10}}(3\hat{i} + \hat{k})\) and \(\vec{b} = \dfrac{1}{7}(2\hat{i} + 3\hat{j} - 6\hat{k})\), then the value of \((2\vec{a} - \vec{b}) \cdot [(\vec{a} \times \vec{b}) \times (\vec{a} + 2\vec{b})]\) is</p>
Step-by-Step Solution
Key Concept: Recognize that (2a - b) is a linear combination of a and b, so it lies in the plane of a and b. Therefore (2a - b) is perpendicular to any vector in the form (a × b) × (anything), making the dot product zero by the properties of scalar triple product.
Step 1: Observe that (2a - b) is a linear combination of vectors a and b. Therefore, (2a - b) lies in the plane spanned by a and b. Step 2: The vector (a × b) is perpendicular to the plane containing a and b. Step 3: The vector (a × b) × (a + 2b) is the cross product of (a × b) with another vector. By properties of the vector triple product: (a × b) × (a + 2b) is perpendicular to (a × b). Step 4: Since (2a - b) lies in the plane of a and b, it is perpendicular to (a × b). Any vector in the span of a and b must be orthogonal to (a × b) × (a + 2b) because this vector is orthogonal to (a × b). Step 5: Therefore: (2a - b) · [(a × b) × (a + 2b)] = 0 ∴ Answer: 0
Correct Answer: D