Vector Algebra
Properties of vector magnitudes and dot products
Grade 12
Question:
<p><strong>a, b and c</strong> are three vectors such that <strong>a</strong> · <strong>a</strong> = <strong>b</strong> · <strong>b</strong> = <strong>c</strong> · <strong>c</strong> = 3 and |<strong>a</strong> − <strong>b</strong>|² + |<strong>b</strong> − <strong>c</strong>|² + |<strong>c</strong> − <strong>a</strong>|² = 27, then</p>
<p>(a) <strong>a, b and c</strong> are necessarily coplanar.</p>
<p>(b) <strong>a, b and c</strong> represent sides of a triangle in magnitude and direction</p>
<p>(c) <strong>a</strong> · <strong>b</strong> + <strong>b</strong> · <strong>c</strong> + <strong>c</strong> · <strong>a</strong> has the least value −9/2</p>
<p>(d) <strong>a, b and c</strong> represent orthogonal triad of vectors</p>
Step-by-Step Solution
Key Concept: Expand the magnitude condition to find the sum of dot products, then use |**a** + **b** + **c**|² ≥ 0 to prove linear dependence and coplanarity.
Step 1: Expand | a − b |^2 + | b − c |^2 + | c − a |^2: \(= 2(|\mathbf{a}|^2 + |\mathbf{b}|^2 + |\mathbf{c}|^2) - 2(\mathbf{a}·\mathbf{b} + \mathbf{b}·\mathbf{c} + \mathbf{c}·\mathbf{a})\) \(= 2(3 + 3 + 3) - 2(\mathbf{a}·\mathbf{b} + \mathbf{b}·\mathbf{c} + \mathbf{c}·\mathbf{a}) = 27\) Step 2: Solving for dot products: \(\mathbf{a}·\mathbf{b} + \mathbf{b}·\mathbf{c} + \mathbf{c}·\mathbf{a} = -\frac{9}{2}\) Step 3: Consider | a + b + c |^2: \(|\mathbf{a} + \mathbf{b} + \mathbf{c}|^2 = 9 - 9 = 0\) Therefore a + b + c = 0 ...(i) Step 4: This means the three vectors form a closed triangle (representing sides in magnitude and direction), hence they are coplanar and a · b + b · c + c · a = −9/2 is the necessary value. ∴ Answers are (a), (b), and (c).
Correct Answer: A, B, C