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.

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?

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 $\left|\overrightarrow{\mathrm{OB}}\right|$?

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.

3v - 4w

Given two pairs of vectors, $a$ and $b$, and $e$ and $f$, if $a\cdot b$ = $0$ and $e\cdot f$ = $0$, what can be concluded about the relationship between each pair of vectors?

In Exercises 1–4, u and v have the same direction. In each exercise: Is u = v? Explain.

Point S lies between points R and T on a straight line. If $RT$ is $10$ centimeters long and $RS$ is $4$ centimeters, what is the length of $ST$?

Determine if the following statement is true or false: Temperature is a vector. 

Determine if the following statement is true or false: Acceleration is a vector. 

Determine if the following statement is true or false: The vectors $v$ and $-v$ point in the same direction.