8. Vectors

Given the line with parametric equations $x=1+2t$, $y=3-t$, $z=4+5t$, find the perpendicular distance from the point $(2,1,0)$ to the line.

Given vectors $u ⃗$ and $v ⃗$, sketch the resultant vector $\(\frac\)12u⃗+v⃗$.

In circle O, if chord $AE$ has length $7$ units and passes through the center O, what is the radius of the circle?

Given the vector $4i-j+8k$, which of the following is a unit vector in the same direction?

Find the magnitude and direction angle for each vector. Round angle measures to the nearest tenth, as necessary.

〈-4, 4√3〉

If you draw all possible diagonals from a single vertex of a regular $decagon$, how many distinct triangles can be formed using that vertex and two other vertices of the $decagon$?

Point A is the center of a circle. If the ratio of the lengths of $\left|\overrightarrow{\mathrm{OA}}\right|$ to $\left|\overrightarrow{\mathrm{OB}}\right|$ is $2:1$, what is the ratio of the areas of the circles with radii $\left|\overrightarrow{\mathrm{OA}}\right|$ and $\left|\overrightarrow{\mathrm{OB}}\right|$?

A plane has an airspeed of 520 mph. The pilot wishes to fly on a bearing of 310°. A wind of 37 mph is blowing from a bearing of 212°. In what direction should the pilot fly, and what will be her ground speed?

Given the vectors $a=\left(3,,,12\right)$ and $b=\left(6,,,9\right)$, find the orthogonal projection of $a$ onto $b$ (denoted as $\mathrm{orth}ab$).

In which diagram are angles $1$ and $2$ vertical angles?

Given two lines $a$ and $d$ that are non-coplanar, which of the following best describes their relationship?

In Exercises 21–38, let u = 2i - 5j, v = -3i + 7j, and w = -i - 6j. Find each specified vector or scalar.

5v

Use the figure to find each vector: u - v. Use vector notation as in Example 4.