Calculus 3: Dot and cross product

Calculate the dot product between vector $\mathbf{a} = (4, 5)$ and the sum of vectors $\mathbf{b} = (3, -6)$ and $\mathbf{c} = (-8, 2)$.

Find the cross product of the vectors $\mathbf{A} = \langle 1, 3, 4 \rangle$ and $\mathbf{B} = \langle 2, 7, -5 \rangle$.

Find the cross product of vectors $\mathbf{a}$ and $\mathbf{b}$, where $\mathbf{a} = 3\mathbf{i} + 5\mathbf{j} - 7\mathbf{k}$ and $\mathbf{b} = 2\mathbf{i} - 6\mathbf{j} + 4\mathbf{k}$.

Using the vector cross product, determine the vector perpendicular to two given initial vectors using the right-hand rule.

Calculate the dot product of $\mathbf{u}$ and $3\mathbf{v}$ by using a shortcut method for scalar multiplication.

Find $w_2$, the vector component of $\mathbf{u}$ orthogonal to $\mathbf{v}$, where $\mathbf{u} = (3, 5)$ and $\mathbf{v} = (2, 4)$.

For vectors $\mathbf{u} = 6\mathbf{i} - 3\mathbf{j} + 9\mathbf{k}$ and $\mathbf{v} = 4\mathbf{i} - \mathbf{j} + 8\mathbf{k}$, find the two components $w_1$ and $w_2$ of vector $\mathbf{u}$, where $w_1$ is the projection of $\mathbf{u}$ onto $\mathbf{v}$ and $w_2$ is the component orthogonal to $\mathbf{v}$.