The figure below shows two vectors $\mathbf{a}$ and $\mathbf{b}$ in $\mathbb{R}^3$ that are in the $xy$-plane. Suppose $\| \mathbf{a} \| = 4$, $\| \mathbf{b} \| = 3$, and the angle between the two vectors is $50$ degrees.

1. The dot product of these vectors is $\mathbf{a \cdot b} =$ . You should be able to determine the sign of $\mathbf{a \cdot b}$ without making any calculations.
2. The cross product $\mathbf{a} \times \mathbf{b}$ points in the same direction as the vector choose i j k - i - j - k . You should be able to determine this without making any calculations.
3. The length of $\mathbf{a} \times \mathbf{b}$ is $\| \mathbf{a} \times \mathbf{b} \| =$ .
4. The cross product $\mathbf{a} \times \mathbf{b}$ is .
5. The area of the triangle formed by $\mathbf{a}$ and $\mathbf{b}$ is .

The figure below shows two vectors $\mathbf{a}$ and $\mathbf{b}$ in $\mathbb{R}^3$ that are in the $xy$-plane. Suppose $\| \mathbf{a} \| = 2$, $\| \mathbf{b} \| = 4$, and the angle between the two vectors is $100$ degrees.

1. The area of the parallelogram formed by $\mathbf{a}$ and $\mathbf{b}$ is .
2. The cross product $\mathbf{a} \times \mathbf{b}$ points in the same direction as the vector choose i j k - i - j - k . You should be able to determine this without making any calculations.
3. The cross product $\mathbf{a} \times \mathbf{b}$ is .
4. The dot product $\mathbf{k} \cdot (\mathbf{a} \times \mathbf{b})$ is .
5. The dot product $(2 \mathbf{i} + 4 \mathbf{j}) \cdot (\mathbf{a} \times \mathbf{b})$ is . Using vector geometry, this should be obvious.