Vector Algebra
Orthogonal Vectors
Grade 12
Question:
<p>If <strong>b</strong> and <strong>c</strong> are orthogonal unit vectors and \(\mathbf{b} \times \mathbf{c} = \mathbf{a}\), then \([\mathbf{a} + \mathbf{b} + \mathbf{c} \mathbf{a} + \mathbf{b} \mathbf{b} + \mathbf{c}]\) is equal to:</p>
Step-by-Step Solution
Key Concept: Recognize that the scalar triple product reduces to the dot product of a vector with itself when the vectors form an orthonormal set.
Given: b and c are orthogonal unit vectors, \(\mathbf{b} \times \mathbf{c} = \mathbf{a}\) Step 1: Expand the scalar triple product \([\mathbf{a} + \mathbf{b} + \mathbf{c} \mathbf{a} + \mathbf{b} \mathbf{b} + \mathbf{c}] = [\mathbf{a} \mathbf{a} + \mathbf{b} \mathbf{b} + \mathbf{c}] + [\mathbf{b} + \mathbf{c} \mathbf{a} + \mathbf{b} \mathbf{b} + \mathbf{c}]\) Step 2: Simplify using properties \(= [\mathbf{a} \mathbf{b} \mathbf{c}] = \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = \mathbf{a} \cdot \mathbf{a} = |\mathbf{a}|^2\) Step 3: Calculate magnitude of a Since | b | = | c | = 1 and they are orthogonal: \(|\mathbf{a}|^2 = |\mathbf{b} \times \mathbf{c}|^2 = 1\) ∴ Answer is 1.
Correct Answer: 1