Vector Algebra
Scalar Triple Product
Grade 12
Question:
<p>If \(|\vec{a}| = 1\), \(|\vec{b}| = 3\) and \(|\vec{c}| = 5\), then the value of \([\vec{a} - \vec{b}, \vec{b} - \vec{c}, \vec{c} - \vec{a}]\) is</p>
<p>(a) 0</p>
<p>(b) 1</p>
<p>(c) -1</p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: Three vectors whose sum is zero are linearly dependent and coplanar, so their scalar triple product must be zero.
Step 1: The scalar triple product \([\vec{a} - \vec{b}, \vec{b} - \vec{c}, \vec{c} - \vec{a}]\) represents the determinant of vectors. Step 2: Notice that: \[(\vec{a} - \vec{b}) + (\vec{b} - \vec{c}) + (\vec{c} - \vec{a}) = \vec{0}\] Step 3: When three vectors sum to zero, they are linearly dependent and therefore coplanar. Step 4: The scalar triple product of coplanar vectors is zero: \[[\vec{a} - \vec{b}, \vec{b} - \vec{c}, \vec{c} - \vec{a}] = 0\] ∴ Answer is (a) 0
Correct Answer: A