Vector Algebra
Dot Product and Linear Equations
Grade None
Question:
<p>We have \(\vec{a} \cdot \vec{b} = 0,\ \vec{b} \cdot \vec{c} = 0,\ \vec{c} \cdot \vec{a} = 0\). Given \(2\lambda + 4 + \mu = 0\) and \(\lambda - 1 + 2\mu = 0\). Find the values of \(\lambda\) and \(\mu\).</p>
<p>\(\lambda = -3,\ \mu = 2\)</p>
<p>\(\lambda = 3,\ \mu = -2\)</p>
<p>\(\lambda = -3,\ \mu = -2\)</p>
<p>\(\lambda = 3,\ \mu = 2\)</p>
Step-by-Step Solution
Key Concept: Three mutually orthogonal vectors satisfy independent linear constraints. Solve the system of two linear equations in two unknowns using substitution or elimination to find unique values of λ and μ.
Step 1: Write the system of equations clearly: Equation (1): 2λ + μ + 4 = 0 Equation (2): λ + 2μ - 1 = 0 Step 2: From equation (2), express λ in terms of μ: λ = 1 - 2μ Step 3: Substitute into equation (1): 2(1 - 2μ) + μ + 4 = 0 2 - 4μ + μ + 4 = 0 6 - 3μ = 0 μ = 2 Step 4: Find λ using λ = 1 - 2μ: λ = 1 - 2(2) = 1 - 4 = -3 Verification: 2(-3) + 2 + 4 = -6 + 6 = 0 ✓ and -3 + 2(2) - 1 = -3 + 4 - 1 = 0 ✓ ∴ Answer: λ = -3, μ = 2
Correct Answer: A