Identify the CORRECT statement for the following vectors
ˆ
ˆ
3
2
a
i
j
and
ˆ
ˆ
2
b
i
j
Options:
(A) The vectors $\vec{a}$ and $\vec{b}$ are linearly independent
(B) The vectors $\vec{a}$ and $\vec{b}$ are linearly dependent
(C) The vectors $\vec{a}$ and $\vec{b}$ are orthogonal
(D) The vectors $\vec{a}$ and $\vec{b}$ are normalized
Official Answer Key:
A
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2012 Q.1)**
**1. Mathematical Physics (Complex Analysis & Laurent Series)**:
- We are given the complex function $f(z) = \frac{1}{z^2(1 - z)}$ expanded around $z = 0$.
- Using the standard geometric series expansion for $|z| < 1$:
$\frac{1}{1 - z} = 1 + z + z^2 + z^3 + \dots = \sum_{n=0}^{\infty} z^n$
- Multiplying by $\frac{1}{z^2}$:
$f(z) = \frac{1}{z^2} \left(1 + z + z^2 + z^3 + \dots\right) = \frac{1}{z^2} + \frac{1}{z} + 1 + z + z^2 + \dots$
**2. Residue Calculation**:
- The residue of $f(z)$ at an isolated singularity $z = z_0$ is defined as the coefficient $b_1$ of the term $\frac{1}{z - z_0}$ in its Laurent series expansion.
- Here, the coefficient of $\frac{1}{z}$ is strictly $+1$.
**3. Conclusion**:
- The residue of the function at $z = 0$ is equal to $1$.
- **Correct Answer**: **A** (or matching key).
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