5.3 3D Vectors

Let [math]\overrightarrow{a}=2\textbf{i}+2\textbf{j}+\textbf{k}, \overrightarrow{b}=-3\textbf{i}+0\textbf{j}+3\textbf{k},[/math] and [math] \overrightarrow{c}=\textbf{i}-1.5\textbf{j}+\textbf{k}[/math]. Find

(a) [math]3\overrightarrow{a}\cdot(\overrightarrow{b}-2\overrightarrow{c})[/math]

(b) [math]\|\overrightarrow{b}+2\overrightarrow{c}\|^2[/math]

(c) [math]\overrightarrow{b}\times\overrightarrow{a}[/math]

(d) [math]\text{proj}_{\vec{a}}\overrightarrow{c}[/math]

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]?

Two vectors [math]\overrightarrow{u}=6\textbf{i}-\textbf{j}+3\textbf{k}[/math] and [math]\overrightarrow{v}=-1\textbf{i}+3\textbf{j}+2\textbf{k}[/math] are

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]?

The figure to the right shows a box. [math]AD=AB=3[/math] and [math]DH=2[/math].

(a) Evaluate [math]\overrightarrow{AF}\cdot\overrightarrow{HA}[/math].

(b) If [math]\overrightarrow{AM}=a\overrightarrow{AB}+b\overrightarrow{AD}+c\overrightarrow{AE}[/math] where [math]a, b[/math], and [math]c[/math] are constants and [math]M[/math] is the midpoint of [math]\overline{CG}[/math], what are [math]a, b[/math], and [math]c[/math]?

(c) What is the volume of [math]E-AFH[/math]?

(d) If the distance between [math]E[/math] and the plane [math]AFH[/math] is [math]p\sqrt{51}[/math] where [math]p[/math] is a rational number, what is the value of [math]p[/math]?

(Triangular inequality) Let [math]\overrightarrow{a}[/math] and [math]\overrightarrow{b}[/math] be two vectors. Which of the following inequalities must be true?

Let [math]\overrightarrow{a}=3\textbf{i}+12\textbf{j}-4\textbf{k}[/math] and [math]\overrightarrow{b}=\textbf{i}-\textbf{j}+\textbf{k}[/math].

(a) How many distinct unit vectors are parallel to [math]\overrightarrow{a}[/math]? If your answer is infinite, write “Inf”.

(b) How many distinct unit vectors are perpendicular to [math]\overrightarrow{b}[/math]? If your answer is infinite, write “Inf”.

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.

Consider three points [math]O(0,0,0), A(3,-2,6)[/math], and [math]B(-2,2,-1)[/math]. If point [math]C[/math] is on segment [math]\overline{OA}[/math] such that [math]\overline{BC}[/math] is perpendicular to [math]\overline{OA}[/math], find the coordinates of [math]C[/math].

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].

Find a vector perpendicular to vectors [math]\overrightarrow{u}=1\textbf{i}+2\textbf{j}+3\textbf{k}[/math] and [math]\overrightarrow{v}=4\textbf{i}+5\textbf{j}+6\textbf{k}[/math].
Note: Each box contains a whole number

Let [math]\overrightarrow{u}=t\textbf{i}+3\textbf{j}+(t-1)\textbf{k}[/math] and [math]\overrightarrow{v}=1\textbf{i}+t\textbf{j}+2t\textbf{k}[/math] be two vectors where [math]t[/math] is a real number. Find [math]t[/math] such that [math]\overrightarrow{u}\cdot\overrightarrow{v}=\|\overrightarrow{u}\|\|\overrightarrow{v}\|[/math]. If there is no [math]t[/math] that satisfies the equation, write “None”

Which of the following statement(s) is/are true?

Which of the following statement(s) is/are true?