If $\vec{a}=6 \vec{i}-5 \vec{j}+2 \vec{k}$ and $\vec{b}=3 \vec{i}+4 \vec{j}-7 \vec{k}$, find $3 \vec{a}-2 \vec{b}$.

Update your journal with what you have learned since the last entry. Include the definition, computational technique, and uses of the dot product, along with its two other names.

For the following problem, use the definition of dot product to find $\vec{a} \cdot \vec{b}$, where $\theta$ is the angle between $\vec{a}$ and $\vec{b}$ when they are placed tail-to-tail.

$|\vec{a}|=29,|\vec{b}|=50$, and $\theta=127^{\circ}$

For the following problem, find the cross product using determinants.

$$(4 \vec{i}-3 \vec{j}-\vec{k}) \times(2 \vec{i}-\vec{j}+\vec{k})$$

For the following problem, draw the position vector on graph paper. Show the circumscribed "box" that makes the vector look three-dimensional. Write the coordinates of the point at the head of the vector.

$$\vec{p}=3 \vec{i}+8 \vec{j}+4 \vec{k}$$

The position vector for a point $(x, y)$ in a coordinate system has a fixed location. It goes from the origin to the point. Given the position vector $\vec p$ for the point $(4,3)$. You can think of this vector as the sum of a vector in the $x$-direction and a vector in the $y$-direction. Sketch the two components of $\vec{P}$.

Find the point on the line in the previous problem for which d = 34.

For the following problem, find the direction cosines and direction angles for the position vector to the given point.

(2,-5,3)

For the following problem, find a particular equation of the plane described.

Perpendicular to $\vec{n}=3 \vec{i}-5 \vec{j}+4 \vec{k}$, containing the point (6,-7,-2)

What is the major difference in meaning between $\vec{a} \cdot \vec{b}$ and $\vec{a} \times \vec{b}$?

Given a cube with one corner at the origin of a three-dimensional coordinate system.

Find the angle the main diagonal makes with a diagonal in a face of the cube.

Write the coordinates of the point 40% of the way from the head of $\vec{a}$ to the head of $\vec{b}$.

Given a cube with one corner at the origin of a three-dimensional coordinate system.

Find the angle the main diagonal of the cube makes with one of the edges of the cube.

Find the position vector for the point 40% of the way from the head of $\vec{a}$ to the head of $\vec{b}$.