To determine which of the given vectors is orthogonal to both $\mathbf{v_1} = (1, -3, 0)$ and $\mathbf{v_2} = (-1, 2, 1)$, we need to find the cross product of $\mathbf{v_1}$ and $\mathbf{v_2}$. The result of this cross product will give us a vector that is orthogonal to both $\mathbf{v_1}$ and $\mathbf{v_2}$. We can then compare the cross product with the given options.

Step 1: Compute the cross product of $\mathbf{v_1}$ and $\mathbf{v_2}$.

The cross product of two vectors $\mathbf{v_1} = (v_{1x}, v_{1y}, v_{1z})$ and $\mathbf{v_2} = (v_{2x}, v_{2y}, v_{2z})$ is given by:

$\mathbf{v_1} \times \mathbf{v_2} = \left( v_{1y}v_{2z} - v_{1z}v_{2y}, v_{1z}v_{2x} - v_{1x}v_{2z}, v_{1x}v_{2y} - v_{1y}v_{2x} \right)$

Substitute the components of $\mathbf{v_1} = (1, -3, 0)$ and $\mathbf{v_2} = (-1, 2, 1)$:

$\mathbf{v_1} \times \mathbf{v_2} = \left( (-3)(1) - (0)(2), (0)(-1) - (1)(1), (1)(2) - (-3)(-1) \right)$

This simplifies to:

$\mathbf{v_1} \times \mathbf{v_2} = \left( -3 - 0, 0 - 1, 2 - 3 \right) = (-3, -1, -1)$

Thus, the vector orthogonal to both $\mathbf{v_1}$ and $\mathbf{v_2}$ is $(-3, -1, -1)$.

Step 2: Compare this with the given options.

We need to check if any of the provided vectors is a scalar multiple of $(-3, -1, -1)$.

• $(2, -2, 6)$ is not a multiple of $(-3, -1, -1)$.
• $(3, 1, 1)$ is a multiple of $(-3, -1, -1)$ because $(3, 1, 1) = -1 \times (-3, -1, -1)$.

Therefore, $(3, 1, 1)$ is the vector orthogonal to both $\mathbf{v_1}$ and $\mathbf{v_2}$.

Final Answer:

The vector $(3, 1, 1)$ is orthogonal to both $\mathbf{v_1}$ and $\mathbf{v_2}$.

Do you want further details or have any questions?

Here are five related questions to deepen your understanding:

1. What is the geometric interpretation of the cross product?
2. How do we find the magnitude of the vector resulting from a cross product?
3. Can two non-parallel vectors in $\mathbb{R}^3$ have more than one orthogonal vector?
4. What is the significance of the scalar triple product of three vectors?
5. How can we use the dot product to verify orthogonality of two vectors?

Tip: The cross product of two vectors gives a vector that is perpendicular to both in 3D space, and its direction is given by the right-hand rule.