Verify if the vectors ⃗a =⃗i+2⃗j,
⃗b = 2⃗i+4⃗j, and ⃗c = −⃗i−2⃗j are linearly
dependent. (13 marks)

To verify if the vectors \(\vec{a} = \hat{i} + 2\hat{j}\), \(\vec{b} = 2\hat{i} + 4\hat{j}\), and \(\vec{c} = -\hat{i} - 2\hat{j}\) are linearly dependent, we need to check if there exists a non-trivial linear combination of these vectors that results in the zero vector.

### Step 1: Express the linear combination
The vectors are linearly dependent if there exist scalars \(x_1\), \(x_2\), and \(x_3\), **not all zero**, such that:

\[
x_1 \vec{a} + x_2 \vec{b} + x_3 \vec{c} = \vec{0}
\]

In component form:

\[
x_1(\hat{i} + 2\hat{j}) + x_2(2\hat{i} + 4\hat{j}) + x_3(-\hat{i} - 2\hat{j}) = \vec{0}
\]

### Step 2: Set up the system of equations
By breaking this into components for \(\hat{i}\) and \(\hat{j}\), we get two equations:

For \(\hat{i}\):
\[
x_1(1) + x_2(2) + x_3(-1) = 0
\]
\[
x_1 + 2x_2 - x_3 = 0 \quad \text{(Equation 1)}
\]

For \(\hat{j}\):
\[
x_1(2) + x_2(4) + x_3(-2) = 0
\]
\[
2x_1 + 4x_2 - 2x_3 = 0 \quad \text{(Equation 2)}
\]

### Step 3: Solve the system of equations
We now solve this system:

#### From Equation 1:
\[
x_1 + 2x_2 - x_3 = 0 \quad \Rightarrow \quad x_1 = x_3 - 2x_2
\]

#### Substitute \(x_1 = x_3 - 2x_2\) into Equation 2:
\[
2(x_3 - 2x_2) + 4x_2 - 2x_3 = 0
\]
Expand:
\[
2x_3 - 4x_2 + 4x_2 - 2x_3 = 0
\]
Simplify:
\[
0 = 0
\]

This equation is always true, so there are infinitely many solutions. This implies that the system is **linearly dependent**.

### Step 4: Conclusion
Since there are infinitely many solutions (not just the trivial solution where all scalars are zero), the vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\) are linearly dependent.

### Final Answer:
Yes, the vectors \(\vec{a} = \hat{i} + 2\hat{j}\), \(\vec{b} = 2\hat{i} + 4\hat{j}\), and \(\vec{c} = -\hat{i} - 2\hat{j}\) are linearly dependent.

---

Do you want further details or have any questions?

Here are five related questions to deepen your understanding:
1. How can we geometrically interpret linearly dependent vectors in two dimensions?
2. Can three linearly dependent vectors form a basis in \(\mathbb{R}^2\)?
3. What conditions must hold for three 2D vectors to be linearly independent?
4. How would this process change if we were working in \(\mathbb{R}^3\) instead?
5. What is the role of the determinant in identifying linear dependence for vectors?

**Tip**: Linearly dependent vectors in 2D are collinear, meaning they lie along the same line.

Verify if the vectors ⃗a =⃗i+2⃗j, ⃗b = 2⃗i+4⃗j, and ⃗c = −⃗i−2⃗j are linearly dependent.

Verify if the vectors ⃗a =⃗i+2⃗j, ⃗b = 2⃗i+4⃗j, and ⃗c = −⃗i−2⃗j are linearly dependent using linear combinations. This solution covers key concepts from linear algebra and vector spaces, suitable for undergraduate-level students.

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