Vector Algebra
Vector Products and Properties
Grade 12
Question:
<p>If $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ are three non-zero vectors, then which of the following statement(s) is/are true?</p><p>(a) $\mathbf{a} \times (\mathbf{b} \times \mathbf{c})$, $\mathbf{b} \times (\mathbf{c} \times \mathbf{a})$, $\mathbf{c} \times (\mathbf{a} \times \mathbf{b})$ form a right handed system.</p><p>(b) $\mathbf{c}$, $(\mathbf{a} \times \mathbf{b}) \times \mathbf{c}$, $\mathbf{a} \times \mathbf{b}$ form a right handed system.</p><p>(c) $\mathbf{a} \cdot \mathbf{b} + \mathbf{b} \cdot \mathbf{c} + \mathbf{c} \cdot \mathbf{a} < 0$, if $\mathbf{a} + \mathbf{b} + \mathbf{c} = \mathbf{0}$</p><p>(d) $(\mathbf{a} \times \mathbf{b}) \cdot (\mathbf{b} \times \mathbf{c}) = -1$, if $\mathbf{a} + \mathbf{b} + \mathbf{c} = \mathbf{0}$</p>
<p>(a)</p>
<p>(b)</p>
<p>(c)</p>
<p>(d)</p>
Step-by-Step Solution
Key Concept: Use properties of vector triple products and the constraint $\mathbf{a} + \mathbf{b} + \mathbf{c} = \mathbf{0}$ to determine which statements are true. The scalar triple product identity and cyclic properties are essential.
Solution: (a) $\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) + \mathbf{b} \times (\mathbf{c} \times \mathbf{a}) + \mathbf{c} \times (\mathbf{a} \times \mathbf{b}) = \mathbf{0}$ ⇒ vectors are coplanar, so do not form RHS. (b) $\mathbf{c}$, $(\mathbf{a} \times \mathbf{b}) \times \mathbf{c}$, $\mathbf{a} \times \mathbf{b}$ form RHS as they are in the same cyclic order. (c) $\mathbf{a} + \mathbf{b} + \mathbf{c} = \mathbf{0}$ ⇒ $|\mathbf{a} + \mathbf{b} + \mathbf{c}|^2 = 0$ $\Rightarrow \mathbf{a}^2 + \mathbf{b}^2 + \mathbf{c}^2 = -2(\mathbf{a} \cdot \mathbf{b} + \mathbf{b} \cdot \mathbf{c} + \mathbf{c} \cdot \mathbf{a})$ $\Rightarrow \mathbf{a} \cdot \mathbf{b} + \mathbf{b} \cdot \mathbf{c} + \mathbf{c} \cdot \mathbf{a} < 0$ (d) From condition (c), when $\mathbf{a} + \mathbf{b} + \mathbf{c} = \mathbf{0}$: $\mathbf{a} \times \mathbf{b} = \mathbf{b} \times \mathbf{c} = \mathbf{c} \times \mathbf{a}$, therefore $(\mathbf{a} \times \mathbf{b}) \cdot (\mathbf{b} \times \mathbf{c}) = -1$ holds.
Correct Answer: b, c, d