Vector Algebra
Linear Dependence of Vectors
Grade 12
Question:
<p>Given three vectors \(\vec{U}=\hat{i}+\hat{j}+\hat{k}\), \(\vec{V}=\hat{i}+\hat{j}-\hat{k}\), \(\vec{W}=\hat{i}-\hat{j}+\hat{k}\). Which of the following hold?</p>
<li>\(\vec{U},\vec{V},\vec{W}\) are linearly independent</li>
<li>\(\vec{U},\vec{V},\vec{W}\) form a basis for \(\mathbb{R}^3\)</li>
<li>\(\vec{U}+\vec{V}+\vec{W}=3\hat{i}+\hat{j}+\hat{k}\)</li>
<li>\(\vec{U},\vec{V},\vec{W}\) are mutually perpendicular</li>
Step-by-Step Solution
Key Concept: Compute det[U,V,W]. If non-zero, they are linearly independent (and form a basis). Check mutual perpendicularity separately.
$\det\begin{pmatrix}1&1&1\\1&1&-1\\1&-1&1\end{pmatrix}=1(1-1)-1(1+1)+1(-1-1)=0-2-2=-4\neq0$.
So U, V, W are linearly independent ✓ (A) and form a basis ✓ (B).
Check perpendicularity: $\vec{U}\cdot\vec{V}=1+1-1=1\neq0$. Not mutually perpendicular ✗ (D).
Answer: AB
Correct Answer: AB