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

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 $decagon$?

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?

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

Refer to vectors a through h below. Make a copy or a sketch of each vector, and then draw a sketch to represent each of the following. For example, find a + e by placing a and e so that their initial points coincide. Then use the parallelogram rule to find the resultant, as shown in the figure on the right.

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a + (b + c)

For each pair of vectors u and v with angle θ between them, sketch the resultant.

|u| = 12, |v| = 20, θ = 27°

For each pair of vectors u and v with angle θ between them, sketch the resultant.

|u| = 20, |v| = 30, θ = 30°