Vector Algebra
Triple Products
Grade 12
Question:
<p>If <strong>a</strong> = <strong>i</strong> + 2<strong>j</strong> − 2<strong>k</strong>, <strong>b</strong> = 2<strong>i</strong> − <strong>j</strong> + <strong>k</strong> and <strong>c</strong> = <strong>i</strong> + 3<strong>j</strong> − <strong>k</strong>, then <strong>a</strong> × (<strong>b</strong> × <strong>c</strong>) is equal to</p>
<p>(a) \(20\mathbf{i} - 3\mathbf{j} + 7\mathbf{k}\)</p>
<p>(b) \(20\mathbf{i} - 3\mathbf{j} - 7\mathbf{k}\)</p>
<p>(c) \(20\mathbf{i} + 3\mathbf{j} - 7\mathbf{k}\)</p>
<p>(d) None of the above</p>
Step-by-Step Solution
Key Concept: Use the vector triple product formula a × (b × c) = b(a·c) - c(a·b) to avoid computing cross products sequentially. This formula directly gives the result without needing to first find b × c.
Step 1: Recall the Vector Triple Product Formula
For vectors $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$, the vector triple product is given by:
$$ \mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = \mathbf{b}(\mathbf{a} \cdot \mathbf{c}) - \mathbf{c}(\mathbf{a} \cdot \mathbf{b}) $$
The given vectors are:
$$ \mathbf{a} = \mathbf{i} + 2\mathbf{j} - 2\mathbf{k} $$
$$ \mathbf{b} = 2\mathbf{i} - \mathbf{j} + \mathbf{k} $$
$$ \mathbf{c} = \mathbf{i} + 3\mathbf{j} - \mathbf{k} $$
Step 2: Calculate $\mathbf{a} \cdot \mathbf{c}$
$$ \mathbf{a} \cdot \mathbf{c} = (1)(1) + (2)(3) + (-2)(-1) = 1 + 6 + 2 = 9 $$
Step 3: Calculate $\mathbf{a} \cdot \mathbf{b}$
$$ \mathbf{a} \cdot \mathbf{b} = (1)(2) + (2)(-1) + (-2)(1) = 2 - 2 - 2 = -2 $$
Step 4: Apply the Vector Triple Product Formula
Substitute the calculated dot products into the formula:
$$ \mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = \mathbf{b}(\mathbf{a} \cdot \mathbf{c}) - \mathbf{c}(\mathbf{a} \cdot \mathbf{b}) $$
$$ \mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = (2\mathbf{i} - \mathbf{j} + \mathbf{k})(9) - (\mathbf{i} + 3\mathbf{j} - \mathbf{k})(-2) $$
$$ \mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = (18\mathbf{i} - 9\mathbf{j} + 9\mathbf{k}) - (-2\mathbf{i} - 6\mathbf{j} + 2\mathbf{k}) $$
$$ \mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = 18\mathbf{i} - 9\mathbf{j} + 9\mathbf{k} + 2\mathbf{i} + 6\mathbf{j} - 2\mathbf{k} $$
$$ \mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = (18+2)\mathbf{i} + (-9+6)\mathbf{j} + (9-2)\mathbf{k} $$
$$ \mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = 20\mathbf{i} - 3\mathbf{j} + 7\mathbf{k} $$
Correct Answer: C