1. **State the problem:** We are given a line defined by the parametric equation $\mathbf{r}(t) = (1,1,1) + t(2,1,-1)$ and three vectors $\mathbf{w} = (0,-1,1)$, $\mathbf{v} = (-2,-2,-6)$, and $\mathbf{u} = (1,1,3)$. We want to analyze these vectors in relation to the line. 2. **Understand the line and vectors:** The line passes through point $P = (1,1,1)$ and has direction vector $\mathbf{d} = (2,1,-1)$. The vectors $\mathbf{w}$, $\mathbf{v}$, and $\mathbf{u}$ are given in standard coordinates. 3. **Check if vectors are parallel to the line's direction:** A vector is parallel to the line if it is a scalar multiple of $\mathbf{d}$. - For $\mathbf{w} = (0,-1,1)$, check if there exists $k$ such that $\mathbf{w} = k\mathbf{d}$: $$0 = 2k, \quad -1 = k, \quad 1 = -k$$ From the first equation, $k=0$. From the second, $k=-1$. Contradiction, so $\mathbf{w}$ is not parallel. - For $\mathbf{v} = (-2,-2,-6)$, check if $\mathbf{v} = k\mathbf{d}$: $$-2 = 2k, \quad -2 = k, \quad -6 = -k$$ From first, $k = -1$. From second, $k = -2$. Contradiction, so $\mathbf{v}$ is not parallel. - For $\mathbf{u} = (1,1,3)$, check if $\mathbf{u} = k\mathbf{d}$: $$1 = 2k, \quad 1 = k, \quad 3 = -k$$ From first, $k=\frac{1}{2}$. From second, $k=1$. Contradiction, so $\mathbf{u}$ is not parallel. 4. **Check if vectors are orthogonal to the direction vector:** Two vectors are orthogonal if their dot product is zero. - $\mathbf{w} \cdot \mathbf{d} = 0\cdot 2 + (-1)\cdot 1 + 1 \cdot (-1) = 0 -1 -1 = -2 \neq 0$ - $\mathbf{v} \cdot \mathbf{d} = (-2)\cdot 2 + (-2)\cdot 1 + (-6)\cdot (-1) = -4 -2 +6 = 0$ - $\mathbf{u} \cdot \mathbf{d} = 1\cdot 2 + 1\cdot 1 + 3\cdot (-1) = 2 +1 -3 = 0$ So $\mathbf{v}$ and $\mathbf{u}$ are orthogonal to the direction vector $\mathbf{d}$, but $\mathbf{w}$ is not. 5. **Summary:** - None of the vectors $\mathbf{w}$, $\mathbf{v}$, or $\mathbf{u}$ are parallel to the line's direction vector. - Vectors $\mathbf{v}$ and $\mathbf{u}$ are orthogonal to the line's direction vector. This analysis helps understand the geometric relationship of these vectors to the line.