Vector Algebra
Vector Addition in Trapezium
Grade 12
Question:
<p>In a trapezium, the vector <span class="math">\(\vec{BC} = \lambda \vec{AD}\)</span>. We will then find that <span class="math">\(\vec{p} = \vec{AC} + \vec{BD}\)</span> is collinear with <span class="math">\(\vec{AD}\)</span>. If <span class="math">\(\vec{p} = \mu \vec{AD}\)</span>, then find the relationship between <span class="math">\(\lambda\)</span> and <span class="math">\(\mu\)</span>.</p>
<p>(a) <span class="math">\(\mu = \lambda + 1\)</span></p>
<p>(b) <span class="math">\(\lambda = \mu + 1\)</span></p>
<p>(c) <span class="math">\(\lambda + \mu = 1\)</span></p>
<p>(d) <span class="math">\(\mu = 2 + \lambda\)</span></p>
Step-by-Step Solution
Key Concept: Express the sum of vectors in terms of collinear vectors and use the property of trapezium to simplify the expression.
Step 1: We have \(\vec{p} = \vec{AC} + \vec{BD} = \vec{AC} + \vec{BC} + \vec{CD}\) Step 2: Substitute \(\vec{BC} = \lambda \vec{AD}\) : \(\vec{p} = \vec{AC} + \lambda \vec{AD} + \vec{CD}\) Step 3: Note that \(\vec{AC} + \vec{CD} = \vec{AD}\) in a trapezium: \(\vec{p} = \lambda \vec{AD} + (\vec{AC} + \vec{CD}) = \lambda \vec{AD} + \vec{AD} = (\lambda + 1)\vec{AD}\) Step 4: Since \(\vec{p} = \mu \vec{AD}\) , we have \(\mu = \lambda + 1\) ∴ Answer is (a).
Correct Answer: A