Question:

<b>Solution:</b> We need to find values of $\lambda$ for which the given system of equations has no solution. <b>Step 1: Interpret the First Equation</b> The first equation is: $$|3\vec{r} - 2\vec{a} - \vec{b}| = |\vec{a} - \vec{b}|$$ This represents the set of all points $\vec{r}$ that lie on a sphere centered at $\frac{2\vec{a} + \vec{b}}{3}$ with radius $\frac{|\vec{a} - \vec{b}|}{3}$. We can rewrite this as: $$\left|3\vec{r} - (2\vec{a} + \vec{b})\right| = |\vec{a} - \vec{b}|$$ $$\left|\vec{r} - \frac{2\vec{a} + \vec{b}}{3}\right| = \frac{|\vec{a} - \vec{b}|}{3}$$ Let $\vec{c} = \frac{2\vec{a} + \vec{b}}{3}$ and $R = \frac{|\vec{a} - \vec{b}|}{3}$. So the first equation represents: $|\vec{r} - \vec{c}| = R$ (a sphere with center $\vec{c}$ and radius $R$). <b>Step 2: Interpret the Second Equation</b> The second equation is: $$[\vec{r} - \lambda\vec{a} - (1-\lambda)\vec{b}] \cdot (\vec{a} - \vec{b}) = 0$$ This can be rewritten as: $$[\vec{r} - (\lambda\vec{a} + (1-\lambda)\vec{b})] \cdot (\vec{a} - \vec{b}) = 0$$ This represents the set of all points $\vec{r}$ that lie on a plane perpendicular to $(\vec{a} - \vec{b})$ and passing through the point $\vec{d} = \lambda\vec{a} + (1-\lambda)\vec{b}$. Note that $\vec{d}$ lies on the line joining $\vec{a}$ and $\vec{b}$ for all values of $\lambda$. <b>Step 3: Condition for No Solution</b> The system has no solution when the sphere and the plane do not intersect. This occurs when the distance from the center of the sphere to the plane is greater than the radius of the sphere. The distance from $\vec{c} = \frac{2\vec{a} + \vec{b}}{3}$ to the plane is: $$d = \frac{|(\vec{c} - \vec{d}) \cdot (\vec{a} - \vec{b})|}{|\vec{a} - \vec{b}|}$$ where $\vec{d} = \lambda\vec{a} + (1-\lambda)\vec{b}$. <b>Step 4: Calculate the Distance</b> $$\vec{c} - \vec{d} = \frac{2\vec{a} + \vec{b}}{3} - \lambda\vec{a} - (1-\lambda)\vec{b}$$ $$= \frac{2\vec{a} + \vec{b}}{3} - \lambda\vec{a} - \vec{b} + \lambda\vec{b}$$ $$= \left(\frac{2}{3} - \lambda\right)\vec{a} + \left(\frac{1}{3} - 1 + \lambda\right)\vec{b}$$ $$= \left(\frac{2}{3} - \lambda\right)\vec{a} + \left(\lambda - \frac{2}{3}\right)\vec{b}$$ $$= \left(\frac{2}{3} - \lambda\right)(\vec{a} - \vec{b})$$ <b>Step 5: Apply the