Vector Algebra
Applications to Geometry
Grade 12
Question:
<p>Given a parallelogram ABCD. If |<strong>AB</strong>| = a, |<strong>AD</strong>| = b and |<strong>AC</strong>| = c, then <strong>DB</strong> · <strong>AB</strong> has the value</p>
<p>(a) \(\frac{3a^2 + b^2 - c^2}{2}\)</p>
<p>(b) \(\frac{a^2 + 3b^2 - c^2}{2}\)</p>
<p>(c) \(\frac{a^2 - b^2 + 3c^2}{2}\)</p>
<p>(d) None of the above</p>
Step-by-Step Solution
Key Concept: Express diagonals and sides of a parallelogram using vector addition, then use the dot product to relate the given magnitudes.
In parallelogram ABCD: AC = AB + AD and DB = AB − AD . Therefore, DB · AB = ( AB − AD ) · AB = | AB |^2 − AD · AB . Using | AC |^2 = | AB + AD |^2 = a^2 + b^2 + 2 AB · AD , we get c^2 = a^2 + b^2 + 2 AB · AD . Thus AB · AD = (c^2 − a^2 − b^2)/2. Therefore, DB · AB = a^2 − (c^2 − a^2 − b^2)/2 = (3a^2 + b^2 − c^2)/2.
Correct Answer: A