Calculus 3

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

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 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 $\mathbf{b} = 2\mathbf{i} - 6\mathbf{j} + 4\mathbf{k}$.

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

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

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.

Find the magnitude squared of vector $\mathbf{v} = (-4, 3)$.

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

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

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

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 $\mathbf{v}$.

Find the vector equation, parametric equations, and symmetric equations for the line that passes through the points $(1, 3, -2)$ and $(4, 1, 5)$.

Given a fixed point $P_0$ with coordinates $(x_0, y_0, z_0)$ and a direction vector $\vec{v}$ in three-dimensional space, find the vector equation of a line that passes through $P_0$ and is parallel to $\vec{v}$.

Find the equation of a plane given the three points P(2, 1, 4), Q(4, -2, 7), and R(5, 3, -2).

Given a point $P_0 = (1, 2, 3)$ and a normal vector $\mathbf{n} = (4, 5, 6)$, find the equation of the plane in component form.

Find a vector equation and parametric equations for the line that passes through the point $(5, 1, 3)$ and is parallel to the vector $\mathbf{v} = (1, 4, -2)$. Then find two other points on the line.

Find the parametric and symmetric equations of a line in space given two points.