<p>If \(AB = O\) for the matrices \(A = \begin{bmatrix} \cos^2\theta & \cos\theta\sin\theta \\ \cos\theta\sin\theta & \sin^2\theta \end{bmatrix}\) and \(B = \begin{bmatrix} \cos^2\phi & \cos\phi\sin\phi \\ \cos\phi\sin\phi & \sin^2\phi \end{bmatrix}\), then \(\theta - \phi\) is ___________ (in degree).</p>
Step-by-Step Solution
Key Concept: Recognize that both A and B are rank-1 matrices (outer products of unit vectors), and AB = O implies their column spaces are orthogonal. This forces the direction vectors [cos θ, sin θ] and [cos φ, sin φ] to be perpendicular.
<p><strong>Step 1:</strong> Recognize the structure of matrices A and B. Both are rank-1 matrices expressible as:</p><p>A = <strong>u</strong><strong>u</strong><sup>T</sup> where <strong>u</strong> = [cos θ, sin θ]<sup>T</sup></p><p>B = <strong>v</strong><strong>v</strong><sup>T</sup> where <strong>v</strong> = [cos φ, sin φ]<sup>T</sup></p><p><strong>Step 2:</strong> Compute AB = (<strong>u</strong><strong>u</strong><sup>T</sup>)(<strong>v</strong><strong>v</strong><sup>T</sup>) = <strong>u</strong>(<strong>u</strong><sup>T</sup><strong>v</strong>)<strong>v</strong><sup>T</sup></p><p>Since both <strong>u</strong> and <strong>v</strong> are non-zero vectors, AB = O requires <strong>u</strong><sup>T</sup><strong>v</strong> = 0</p><p><strong>Step 3:</strong> Calculate the dot product:</p><p><strong>u</strong><sup>T</sup><strong>v</strong> = cos θ cos φ + sin θ sin φ = cos(θ - φ) = 0</p><p><strong>Step 4:</strong> Solve for θ - φ:</p><p>cos(θ - φ) = 0</p><p>θ - φ = 90° or 270° (or ±90° + 360°n)</p><p>∴ Answer: <strong>90° (or equivalently ±90°)</strong></p>
Correct Answer: 90