Vectors: Linear Independence Check

We are given two vectors: $ \vec{a} = 3\hat{i} + 2\hat{j} $ $ \vec{b} = \hat{i} + 2\hat{j} $ We need to determine the correct statement about these vectors.

Linear Independence Determination

Two vectors $\vec{a}$ and $\vec{b}$ are linearly dependent if one is a scalar multiple of the other, meaning $\vec{a} = k\vec{b}$ for some scalar $k$. Otherwise, they are linearly independent.

Let's check if $\vec{a} = k\vec{b}$: $ 3\hat{i} + 2\hat{j} = k(\hat{i} + 2\hat{j}) $ $ 3\hat{i} + 2\hat{j} = k\hat{i} + 2k\hat{j} $ Comparing the coefficients of $\hat{i}$ and $\hat{j}$:

• For $\hat{i}$: $3 = k$
• For $\hat{j}$: $2 = 2k \implies k = 1$

Since we get different values for $k$ ($3 \neq 1$), one vector is not a scalar multiple of the other. Therefore, the vectors $\vec{a}$ and $\vec{b}$ are linearly independent.

Orthogonality Check

Vectors are orthogonal if their dot product is zero ($\vec{a} \cdot \vec{b} = 0$). $ \vec{a} \cdot \vec{b} = (3)(1) + (2)(2) = 3 + 4 = 7 $ Since $7 \neq 0$, the vectors are not orthogonal.

Normalization Check

A vector is normalized if its magnitude is 1.

• Magnitude of $\vec{a}$: $|\vec{a}| = \sqrt{3^2 + 2^2} = \sqrt{9 + 4} = \sqrt{13} \neq 1$
• Magnitude of $\vec{b}$: $|\vec{b}| = \sqrt{1^2 + 2^2} = \sqrt{1 + 4} = \sqrt{5} \neq 1$

Neither vector is normalized.

Conclusion

Based on the analysis, the only correct statement is that the vectors $\vec{a}$ and $\vec{b}$ are linearly independent.