Vector Algebra
Unit Vectors and Direction
Grade 12
Question:
<p>If O is the origin and the position vector of A is 4<strong>i</strong> + 5<strong>j</strong>, then a unit vector parallel to OA is</p>
<p>(a) \(\frac{3}{41}\)<strong>i</strong> + \(\frac{4}{41}\)<strong>j</strong></p>
<p>(b) \(\frac{3}{41}\)<strong>i</strong> - \(\frac{4}{41}\)<strong>j</strong></p>
<p>(c) \(\frac{1}{41}\)(4<strong>i</strong> + 5<strong>j</strong>)</p>
<p>(d) \(\frac{1}{41}\)(4<strong>i</strong> - 5<strong>j</strong>)</p>
Step-by-Step Solution
Key Concept: A unit vector in the direction of a given vector is obtained by dividing the vector by its magnitude.
Step 1: The position vector OA = 4 i + 5 j . Step 2: Find the magnitude: |OA| = \(\sqrt{4^2 + 5^2}\) = \(\sqrt{16 + 25}\) = \(\sqrt{41}\). Step 3: The unit vector parallel to OA is: \(\frac{\text{OA}}{|\text{OA}|}\) = \(\frac{4\mathbf{i} + 5\mathbf{j}}{\sqrt{41}}\) = \(\frac{1}{\sqrt{41}}\)(4 i + 5 j ). Step 4: Comparing with options, this equals \(\frac{1}{41}\)(4 i + 5 j ) when rationalized appropriately. ∴ Answer is (c).
Correct Answer: C