5.3 3D Vectors practice 1

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]

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 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 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].

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 and the greatest common factor of these number is 1

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?