Vector Algebra
Vector Magnitude and Properties
Grade 12
Question:
<p>For a non-zero vector \(\mathbf{a}\), the set of real numbers satisfying \(|(5-x)\mathbf{a}| < |2\mathbf{a}|\) consists of all \(x\) such that</p>
<p>(a) \(0 < x < 3\)</p>
<p>(b) \(3 < x < 7\)</p>
<p>(c) \(-7 < x < -3\)</p>
<p>(d) \(-7 < x < 3\)</p>
Step-by-Step Solution
Key Concept: Use the property of vector magnitude |kv| = |k||v|, then solve the resulting linear inequality.
Step 1: We have \(|(5-x)\mathbf{a}| < |2\mathbf{a}|\) Step 2: Using the property \(|k\mathbf{a}| = |k||\mathbf{a}|\): \[|5-x||\mathbf{a}| < 2|\mathbf{a}|\] Step 3: Since \(\mathbf{a}\) is non-zero, \(|\mathbf{a}| > 0\), so we can divide both sides: \[|5-x| < 2\] Step 4: This absolute value inequality gives: \[-2 < 5-x < 2\] Step 5: Solving for \(x\): \[-2-5 < -x < 2-5\] \[-7 < -x < -3\] \[3 < x < 7\] ∴ Answer is (b).
Correct Answer: b