Calculus 3: Dot and cross product

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

Calculate the dot product of vectors $\mathbf{a} = (3, -4, 7)$ and $\mathbf{b} = (5, 2, -3)$.

Calculate the dot product of $a$ and $b$ times vector $a$, where $\mathbf{a} = (2, 3)$ and $\mathbf{b} = (5, -4)$.

Calculate the dot product between vector $b$ and $3a$, where $\mathbf{a} = (2, 3)$ and $\mathbf{b} = (5, -4)$.

Given the magnitudes of vectors $\mathbf{a}$ and $\mathbf{b}$ as 15 and 10 respectively, and the angle between them is 30 degrees, calculate the dot product of the two vectors.

Find the cross product of vectors $\mathbf{a}$ and $\mathbf{b}$, where $\mathbf{a} = 5 \mathbf{i} - 4 \mathbf{j} + 3 \mathbf{k}$ and $\mathbf{b} = -7 \mathbf{i} + 2 \mathbf{j} - 8 \mathbf{k}$.

Compute the dot product of vectors $\mathbf{u} = (3, 12)$ and $\mathbf{v} = (-4, 3)$.

Compute the dot product of vector $\mathbf{u} = (3, 12)$ with itself.

Calculate $(\mathbf{u} \cdot \mathbf{v}) \cdot \mathbf{v}$.

Find $w_1$, the projection of $\mathbf{u}$ onto $\mathbf{v}$, where $\mathbf{u} = (3, 5)$ and $\mathbf{v} = (2, 4)$.