Vector Algebra
Linear Dependence of Vectors
Grade 12
Question:
<p>Given vectors \(\vec{V_1}=a\hat{i}+b\hat{j}+c\hat{k}\), \(\vec{V_2}=b\hat{i}+c\hat{j}+a\hat{k}\), \(\vec{V_3}=c\hat{i}+a\hat{j}+b\hat{k}\). In which of the following conditions are \(\vec{V_1},\vec{V_2},\vec{V_3}\) linearly independent?</p>
<li>\(a+b+c=0\) and \(a^2+b^2+c^2\neq ab+bc+ca\)</li>
<li>\(a+b+c\neq0\) and \(a^2+b^2+c^2=ab+bc+ca\)</li>
<li>\(a=b=c\neq0\)</li>
<li>\(a+b+c\neq0\) and \(a^2+b^2+c^2\neq ab+bc+ca\)</li>
Step-by-Step Solution
Key Concept: Compute det[V_1,V_2,V_3] = a^3+b^3+c^3-3abc = (a+b+c)(a^2+b^2+c^2-ab-bc-ca). For independence, this \neq 0.
The determinant of the matrix formed by $V_1,V_2,V_3$ as rows:
$\det = \begin{vmatrix}a&b&c\\b&c&a\\c&a&b\end{vmatrix} = a^3+b^3+c^3-3abc$
Factor: $a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-bc-ca)$.
For linear independence, $\det\neq0$, which requires both:
$a+b+c\neq0$ AND $a^2+b^2+c^2-ab-bc-ca\neq0$ (i.e., $a^2+b^2+c^2\neq ab+bc+ca$).
Answer: (D)
Correct Answer: D