Let vec{a} = 2hat{i} + 3hat{j} - hat{k} and v | vedclass

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(B) Given that the vector $\vec{c}$ is parallel to the plane containing $\vec{a}$ and $\vec{b}$,it implies that $\vec{c}$ is perpendicular to the norm

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(B) Given that the vector $\vec{c}$ is parallel to the plane containing $\vec{a}$ and $\vec{b}$,it implies that $\vec{c}$ is perpendicular to the normal vector of the plane,which is $(\vec{a} 	imes \vec{b})$.Therefore,the scalar triple product of $\vec{a}, \vec{b},$ and $\vec{c}$ must be zero:$[\vec{a} \, \vec{b} \, \vec{c}] = 0$This is equivalent to the determinant:$\begin{vmatrix} 2 & 3 & -1 \ 1 & -2 & 3 \ \lambda & 1 & 2\lambda - 1 \end{vmatrix} = 0$Expanding the determinant along the first row:$2[(-2)(2\lambda - 1) - 3(1)] - 3[1(2\lambda - 1) - 3(\lambda)] - 1[1(1) - (-2)(\lambda)] = 0$$2[-4\lambda + 2 - 3] - 3[2\lambda - 1 - 3\lambda] - 1[1 + 2\lambda] = 0$$2[-4\lambda - 1] - 3[-\lambda - 1] - 1 - 2\lambda = 0$$-8\lambda - 2 + 3\lambda + 3 - 1 - 2\lambda = 0$$-7\lambda = 0$$\lambda = 0$

Let $\vec{a} = 2\hat{i} + 3\hat{j} - \hat{k}$ and $\vec{b} = \hat{i} - 2\hat{j} + 3\hat{k}$. For what value of $\lambda$ is the vector $\vec{c} = \lambda\hat{i} + \hat{j} + (2\lambda - 1)\hat{k}$ parallel to the plane containing $\vec{a}$ and $\vec{b}$?

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