Vector Algebra
Cross Product of Vectors
Grade 12
Question:
<p>If the vectors \(\vec{a}\), \(\vec{b}\) and \(\vec{c}\) from the sides \(BC\), \(CA\) and \(AB\), respectively, of a triangle \(ABC\), then</p>
<p>\(\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{c} = \vec{c} \cdot \vec{b} = 0\)</p>
<p>\(\vec{a} \times \vec{b} = \vec{b} \times \vec{c} = \vec{c} \times \vec{a}\)</p>
<p>(option 3 not fully visible)</p>
<p>(option 4 not fully visible)</p>
Step-by-Step Solution
Key Concept: The vectors forming the sides of a triangle sum to zero because traversing the complete triangle returns you to the starting point. This fundamental property arises from the closed-loop nature of any triangle.
Step 1: Identify the vectors along triangle sides with proper direction. Step 2: Vector a⃗ = BC (from B to C) Vector b⃗ = CA (from C to A) Vector c⃗ = AB (from A to B) Step 3: Verify closure: Starting from point B, moving along BC gets you to C, then along CA gets you to A, then along AB returns you to B. Step 4: Mathematically: BC + CA + AB represents the complete closed path. Following the path: B → (BC) → C → (CA) → A → (AB) → B This gives: a⃗ + b⃗ + c⃗ = 0⃗ ∴ Answer: a⃗ + b⃗ + c⃗ = 0
Correct Answer: B