Vector Algebra
Scalar Triple Product
Grade 12
Question:
<p>Let \(\vec{a} = \hat{i} - \hat{k}\), \(\vec{b} = x\hat{i} + \hat{j} + (1-x)\hat{k}\) and \(\vec{c} = y\hat{i} + x\hat{j} + (1+x-y)\hat{k}\). Then \([\vec{a}, \vec{b}, \vec{c}]\) depends on</p>
<p>only \(y\)</p>
<p>only \(x\)</p>
<p>both \(x\) and \(y\)</p>
<p>neither \(x\) nor \(y\)</p>
Step-by-Step Solution
Key Concept: The scalar triple product [a,b,c] = a·(b×c) depends only on variables that cannot be eliminated through the determinant calculation. Compute the determinant and identify which variables survive as irreducible parameters.
Step 1: Set up the scalar triple product as a determinant: [a,b,c] = | 1 0 -1 | | x 1 1-x | | y x 1+x-y | Step 2: Expand along the first row: = 1·| 1 1-x | - 0 + (-1)·| x 1 | | x 1+x-y | | y x | Step 3: Calculate the 2×2 determinants: First: 1(1+x-y) - (1-x)x = 1+x-y-x+x^2 = 1+x^2-y Second: x·x - 1·y = x^2-y Step 4: Substitute back: [a,b,c] = (1+x^2-y) - (x^2-y) = 1+x^2-y-x^2+y = 1 Step 5: The scalar triple product equals 1, which is a constant that depends on neither x nor y . ∴ Answer: D (depends on neither x nor y, or is a constant)
Correct Answer: D