The vector that is parallel to the vector 2 h | vedclass

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(C) Let the required vector be $\vec{v} = a \hat{i} + b \hat{j} + c \hat{k}$.Since $\vec{v}$ is coplanar with $\vec{u_1} = \hat{i} + \hat{j}$ and $\ve

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$\hat{i} - \hat{j} - 2 \hat{k}$

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(C) Let the required vector be $\vec{v} = a \hat{i} + b \hat{j} + c \hat{k}$.Since $\vec{v}$ is coplanar with $\vec{u_1} = \hat{i} + \hat{j}$ and $\vec{u_2} = \hat{j} + \hat{k}$, it must be a linear combination of $\vec{u_1}$ and $\vec{u_2}$.Thus, $a \hat{i} + b \hat{j} + c \hat{k} = \lambda(\hat{i} + \hat{j}) + \mu(\hat{j} + \hat{k}) = \lambda \hat{i} + (\lambda + \mu) \hat{j} + \mu \hat{k}$.Comparing coefficients, we get $a = \lambda$, $c = \mu$, and $b = \lambda + \mu$. Substituting $\lambda$ and $\mu$, we get $b = a + c$.The vector $\vec{v}$ is parallel to $2 \hat{i} - 2 \hat{j} - 4 \hat{k}$, so $\vec{v} = k(2 \hat{i} - 2 \hat{j} - 4 \hat{k})$.For option $C$, $\vec{v} = \hat{i} - \hat{j} - 2 \hat{k}$, we have $a = 1, b = -1, c = -2$.Checking the condition $b = a + c$: $1 + (-2) = -1$, which matches $b$.Thus, the vector $\hat{i} - \hat{j} - 2 \hat{k}$ is the correct answer.

The vector that is parallel to the vector $2 \hat{i} - 2 \hat{j} - 4 \hat{k}$ and coplanar with the vectors $\hat{i} + \hat{j}$ and $\hat{j} + \hat{k}$ is

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