5.3 3D Vectors (in class)

If the area of the triangle with vertices [math]A(6,0,6), B(2,8,6)[/math], and [math]C(6,4,5)[/math] is [math]\sqrt{a}[/math] where [math]a[/math] is an integer, what is [math]a[/math]?

The figure shows a regular tetrahedron [math]D-ABC[/math] (the four faces are equilateral triangles). [math]E[/math] is the midpoint of [math]\overline{BC}[/math] and [math]O[/math] is the center of triangle [math]ABC[/math]. Suppose [math]DA=a[/math].

(a) Find [math]\cos\theta[/math] where [math]\theta[/math] is the angle between [math]\overline{DE}[/math] and [math]\overline{EA}[/math].

(b) If [math]DO=\dfrac{\sqrt{p}}{3}a[/math] where [math]p[/math] is an integer, what is [math]p[/math]?

(c) If the volume of [math]D-ABC[/math] is [math]\dfrac{\sqrt{2}}{r}a^3[/math] where [math]r[/math] is an integer, what is the value of [math]r[/math]?

Consider a plane [math]E[/math], a line [math]L[/math], and three points [math]A, B[/math], and [math]C[/math] in [math]\mathbb{R}^3[/math]. [math]A[/math] is not on [math]E[/math] while [math]B, C[/math] and [math]L[/math] are. [math]L[/math] passes through [math]C[/math] but not [math]B[/math]. Given that [math]\overline{AB}\perp E[/math] and [math]\overline{BC}\perp L[/math]. [math]\overline{AC}[/math] is

(Direction cosine) Consider a point [math]A(6,3,2)[/math]. Let [math]\alpha, \beta, \gamma[/math] be acute angles between [math]\overrightarrow{OA}[/math] and the [math]x[/math]-axis, [math]y[/math]-axis, and [math]z[/math]-axis respectively.

Let [math]\overrightarrow{a}=3\textbf{i}+4\textbf{k}, \overrightarrow{b}=-5\textbf{i}+12\textbf{j}[/math] and [math]\overrightarrow{c}=5\textbf{i}-3\textbf{j}+4\textbf{k}[/math] be three vectors. Find

(a) [math]\cos\theta[/math] where [math]\theta[/math] is the angle between [math]\overrightarrow{a}[/math] and [math]\overrightarrow{b}[/math].

(b) the area of the triangle constructed by [math]\overrightarrow{a}[/math] and [math]\overrightarrow{c}[/math].

(c) [math]4\overrightarrow{a}+3(\overrightarrow{b}-\overrightarrow{c})[/math].