Vector Algebra
Unit vector perpendicular to given vectors
Grade 12
Question:
<p>Let \(\vec{u} = \hat{i}+\hat{j}\), \(\vec{v} = \hat{i}-\hat{j}\) and \(\vec{w} = \hat{i}+2\hat{j}+3\hat{k}\). If \(\hat{n}\) is unit vector such that \(\vec{u}\cdot\hat{n} = 0\) and \(\vec{v}\cdot\hat{n} = 0\), then \(|\vec{w}\cdot\hat{n}|\) is equal to</p>
<p>0</p>
<p>1</p>
<p>2</p>
<p>3</p>
Step-by-Step Solution
Key Concept: Find the unit vector perpendicular to both u and v using the cross product, then compute its dot product with w. The vector perpendicular to two vectors in a plane lies along their cross product direction.
Step 1: Since n̂ is perpendicular to both u and v , it must be parallel to u × v . Step 2: Calculate u × v : u × v = (î + ĵ) × (î − ĵ) = î(−1) − ĵ(−1) + k̂(−1−1) = −î + ĵ − 2k̂ Step 3: Find the magnitude: | u × v | = √(1 + 1 + 4) = √6 Step 4: The unit vector is n̂ = ±(−î + ĵ − 2k̂)/√6 Step 5: Calculate w · n̂ : w · n̂ = (î + 2ĵ + 3k̂) · (±(−î + ĵ − 2k̂)/√6) = ±(−1 + 2 − 6)/√6 = ∓5/√6 Step 6: | w · n̂ | = 5/√6 = (5√6)/6 ∴ Answer: D
Correct Answer: D