Vector Algebra
Dot product and magnitude of vectors
Grade 12
Question:
<p>We have <br/>
\(|\hat{x}+\hat{y}|^2 + |\hat{y}+\hat{z}|^2 + |\hat{z}+\hat{x}|^2\)<br/>
where \(\hat{x}, \hat{y}, \hat{z}\) are unit vectors making angles \(\alpha, \beta, \gamma\) with each other. Find the minimum value of \(|\hat{x}+\hat{y}|^2 + |\hat{y}+\hat{z}|^2 + |\hat{z}+\hat{x}|^2\).</p>
<p>3</p>
<p>2</p>
<p>1</p>
<p>0</p>
Step-by-Step Solution
Key Concept: Expand each squared magnitude using the dot product formula |a+b|² = |a|² + |b|² + 2(a·b), then use the constraint that unit vectors satisfy a·b = cos(angle between them) to minimize the sum of cosines of the three pairwise angles.
Step 1: Expand using magnitude formula Since |û|^2 = û·û = 1 for unit vectors: |x̂+ŷ|^2 = |x̂|^2 + |ŷ|^2 + 2(x̂·ŷ) = 1 + 1 + 2cos α = 2(1 + cos α) |ŷ+ẑ|^2 = 2(1 + cos β) |ẑ+x̂|^2 = 2(1 + cos γ) Step 2: Sum the expressions S = 2(1 + cos α) + 2(1 + cos β) + 2(1 + cos γ) S = 6 + 2(cos α + cos β + cos γ) Step 3: Minimize using constraint For three unit vectors in 3D space to exist, they must satisfy closure and geometric constraints. The sum is minimized when cos α + cos β + cos γ is minimized. This occurs when the three vectors are mutually equally spaced. By symmetry and geometric constraint, when α = β = γ = 120°: cos 120° = -1/2 Step 4: Calculate minimum S<sub>min</sub> = 6 + 2(-1/2 - 1/2 - 1/2) = 6 + 2(-3/2) = 6 - 3 = 3 ∴ Answer: 3
Correct Answer: A