Math Snap

STEP 1

1. Vectors $\mathbf{a}$ and $\mathbf{b}$ are given in component form.
2. The cross product of two vectors results in a vector that is perpendicular to both.
3. The cross product of two identical vectors is the zero vector.

STEP 2

1. Identify the components of vectors $\mathbf{a}$ and $\mathbf{b}$.
2. Use the formula for the cross product of two vectors.
3. Compute the cross product $\mathbf{a} \times \mathbf{b}$.

STEP 3

Identify the components of vectors $\mathbf{a}$ and $\mathbf{b}$:
$$\mathbf{a} = \mathbf{i} + \mathbf{j} + 3\mathbf{k}$$ $$\mathbf{b} = \mathbf{i} + \mathbf{j} + 3\mathbf{k}$$

STEP 4

Recall the formula for the cross product of two vectors $\mathbf{u} = u_1 \mathbf{i} + u_2 \mathbf{j} + u_3 \mathbf{k}$ and $\mathbf{v} = v_1 \mathbf{i} + v_2 \mathbf{j} + v_3 \mathbf{k}$:
$$\mathbf{u} \times \mathbf{v} = (u_2 v_3 - u_3 v_2) \mathbf{i} - (u_1 v_3 - u_3 v_1) \mathbf{j} + (u_1 v_2 - u_2 v_1) \mathbf{k}$$

Apply the cross product formula to vectors $\mathbf{a}$ and $\mathbf{b}$:
Since $\mathbf{a} = \mathbf{b}$, the cross product $\mathbf{a} \times \mathbf{b}$ is:
$$\mathbf{a} \times \mathbf{b} = (1 \cdot 3 - 3 \cdot 1) \mathbf{i} - (1 \cdot 3 - 3 \cdot 1) \mathbf{j} + (1 \cdot 1 - 1 \cdot 1) \mathbf{k}$$ Simplifying each term:
$$= (3 - 3) \mathbf{i} - (3 - 3) \mathbf{j} + (1 - 1) \mathbf{k}$$ $$= 0 \mathbf{i} + 0 \mathbf{j} + 0 \mathbf{k}$$ The cross product $\mathbf{a} \times \mathbf{b}$ is:
$$\boxed{0 \mathbf{i} + 0 \mathbf{j} + 0 \mathbf{k}}$$