8. Vectors

Let $MNOP$ be a square with vertices in order. If $t$ and $f$ are the magnitudes of the vectors $\overrightarrow{M}\overrightarrow{N}$ and $\overrightarrow{N}\overrightarrow{O}$ respectively, what are the values of $t$ and $f$ if the side length of the square is $1$? $t$ = $f$ =

In the context of geometric vectors, which of the following best describes orthogonal lines?

Given that segment $TQ$ is $26$ units long, what is the length of $QV$ if $TQ$ and $QV$ are collinear and $TQ$ is a part of $TV$ where $TV$ is $32$ units long?

Given the vectors $u=(3,2)$ and $v=(1,5)$, which of the following lists the correct components for $u$, $v$, $u+v$, and $u-v$ in that order?

Triangle xyz is dilated by a scale factor of $3$ centered at the origin. If the original length of side $xy$ is $5$ units, what is the length of side $xy$ after the transformation?

Which of the following pairs of $\mathrm{triangles}$ can be mapped onto each other using two reflections?

Given the points $(3,5)$ and $(-3,-5)$, which points are reflections of each other across both axes?

Which of the following is one of the basic ways to represent three-dimensional space in trigonometry?

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?