5.2 Inner product

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

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

(b) [math]\|\overrightarrow{a}+\overrightarrow{b}\|[/math]

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

(d) [math]\cos\theta=p\sqrt{2}[/math] where [math]p[/math] is a rational number and [math]\theta[/math] is the angle between [math]\overrightarrow{a}[/math] and [math]\overrightarrow{b}[/math], what is the value of [math]p[/math]?

Let [math]ABCD[/math] be a parallelogram. Given that [math]AB=6[/math] and [math]BC=5[/math], find [math]\overrightarrow{AC}\cdot\overrightarrow{BD}[/math].

Consider a regular hexagon [math]ABCDEF[/math]. Given that [math]AB=1[/math], find

(a) [math]\overrightarrow{AB}\cdot\overrightarrow{AF}[/math]

(b) [math]\overrightarrow{AB}\cdot\overrightarrow{AC}[/math]

(c) [math]\overrightarrow{CB}\cdot\overrightarrow{DF}[/math]

(d) [math]\overrightarrow{AB}\cdot\overrightarrow{BC}[/math]

(e) [math]\overrightarrow{CF}\cdot\overrightarrow{BE}[/math]

Let [math]A(1,3), B(-2,-1)[/math], and [math]C(0,5)[/math] be three points.

(a) Find [math]\overrightarrow{AB}[/math] and [math]\overrightarrow{AC}[/math].

(b) If [math]\cos\theta=\dfrac{a}{\sqrt{b}}[/math] where [math]a[/math] and [math]b[/math] are positive integers and [math]\theta[/math] is the angle between [math]\overrightarrow{AB}[/math] and [math]\overrightarrow{AC}[/math], what is the value of [math]\dfrac{a}{b}[/math]?

(c) Find the area of triangle [math]ABC[/math].

(d) Find the unit vectors [math]\overrightarrow{u}[/math] parallel to [math]\overrightarrow{AB}[/math].

(e) If [math]\dfrac{1}{\sqrt{p}}(q\textbf{i}-r\textbf{j})[/math] is a unit vector perpendicular to [math]\overrightarrow{AC}[/math] where [math]p, q[/math], and [math]r[/math] are positive integers, what is the value of [math]p+q+r[/math]?

Given that [math]\overrightarrow{a}=3\textbf{i}+4\textbf{j}[/math] and [math]\overrightarrow{b}=-2\textbf{i}+\textbf{j}[/math], find

(a) [math]\text{proj}_{\vec{b}}\vec{a}[/math]

(b) [math]\text{proj}_{\vec{a}}\vec{b}[/math]

Let [math]\overrightarrow{a}[/math] and [math]\overrightarrow{b}[/math] be nonzero and nonparallel vectors. Suppose [math]\theta[/math] is the angle between [math]\overrightarrow{a}[/math] and [math]\overrightarrow{b}[/math]. Which of the following statements is/are true?

Let [math]A(3,4)[/math] and [math]B(-15,8)[/math] be two points.

(a) Find [math]\cos\angle AOB[/math] where [math]O[/math] is the origin.

(b) Find the area of [math]AOB[/math].

(c) In triangle [math]AOB[/math], point [math]C[/math] is on [math]\overline{OB}[/math]. If [math]\overline{AC}\perp\overline {OC}[/math], find the coordinates of [math]C[/math].

Consider a parallelogram formed by two vectors [math]\overrightarrow{u}=a\textbf{i}+b\textbf{j}[/math] and [math]\overrightarrow{v}=c\textbf{i}+d\textbf{j}[/math]

(a) Find the area of the parallelogram in terms of [math]a, b, c[/math], and [math]d[/math].

(b) Use the formula in (a) to find the area of the triangle with vertices [math](1,2), (2,5)[/math], and [math](3,4)[/math].

Let [math]\overrightarrow{u}=3\textbf{i}-4\textbf{j}[/math] and [math]\overrightarrow{v}=-1\textbf{i}+6\textbf{j}[/math] be two vectors. Find the constant [math]t[/math] such that [math]\overrightarrow{u}+t\overrightarrow{v}[/math] is perpendicular to [math]\overrightarrow{v}[/math].

Let [math]A(1,-3), B(-3,-1)[/math], and [math]C(3,-2)[/math] be three points.

(a) Find [math]\overrightarrow{AB}[/math] and [math]\overrightarrow{AC}[/math].

(b) Find [math]2\overrightarrow{AB}+\overrightarrow{AC}[/math].

(c) Find [math]\cos\theta[/math] where [math]\theta[/math] is the angle between [math]\overrightarrow{AB}[/math] and [math]\overrightarrow{AC}[/math]

(d) Let [math]D[/math] be a point. If [math]ABCD[/math] is a parallelogram, find the coordinates of [math]D[/math].

(e) Find the area of [math]ABCD[/math].

(f) Find [math]\text{proj}_{\overrightarrow{AC}}\overrightarrow{AB}[/math].

(g) Find [math]\|\text{proj}_{\overrightarrow{AB}}\overrightarrow{AC}\|^2[/math].

(Calculator) Consider vectors [math]\overrightarrow{a}=3\textbf{i}-4\textbf{j}, \overrightarrow{b}=\textbf{i}-\textbf{j}[/math] and [math]\overrightarrow{c}=-7\textbf{i}+24\textbf{j}[/math].

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

(b) The magnitude and direction angle of [math]\overrightarrow{a}+\overrightarrow{b}[/math].

(c) Let [math]\overrightarrow{u}[/math] be a unit vector with direction angle [math]\theta[/math] where [math]\theta[/math] is in the second quadrant. If [math]\overrightarrow{u}[/math] is parallel to [math]\overrightarrow{c}[/math], what is [math]\overrightarrow{u}[/math]?

(d) Let [math]\overrightarrow{v}[/math] be a unit vector with direction angle [math]\theta[/math] where [math]\theta[/math] is in the third quadrant. If [math]\overrightarrow{v}[/math] is perpendicular to [math]\overrightarrow{b}[/math], what is [math]\overrightarrow{v}[/math]?

(e) What is the angle between [math]\overrightarrow{a}+\overrightarrow{b}[/math] and [math]\overrightarrow{a}-\overrightarrow{b}[/math]?

Let [math]\overrightarrow{a}[/math] and [math]\overrightarrow{b}[/math] be unit vectors. If [math]\overrightarrow{a}\cdot\overrightarrow{b}=\dfrac{1}{4}[/math], find [math]\|\overrightarrow{a}-2\overrightarrow{b}\|[/math].

Let [math]ABC[/math] be a triangle satisfying [math]AB=3, BC=\sqrt{7}[/math], and [math]CA=2[/math]. Let [math]O[/math] be its circumcenter.

(a) [math]\overrightarrow{AO}\cdot\overrightarrow{AB}=[/math]
(A) [math]\dfrac{1}{2}\|\overrightarrow{AB}\|^2[/math]
(B) [math]\dfrac{1}{2}\|\overrightarrow{AC}\|^2[/math]
(C) [math]\dfrac{1}{2}\|\overrightarrow{AO}\|^2[/math]
(D) [math]\dfrac{1}{2}\|\overrightarrow{BC}\|^2[/math]

(b) [math]\overrightarrow{AO}\cdot\overrightarrow{AC}=[/math]
(A) [math]\dfrac{1}{2}\|\overrightarrow{AB}\|^2[/math]
(B) [math]\dfrac{1}{2}\|\overrightarrow{AC}\|^2[/math]
(C) [math]\dfrac{1}{2}\|\overrightarrow{AO}\|^2[/math]
(D) [math]\dfrac{1}{2}\|\overrightarrow{BC}\|^2[/math]

(c) Find constants [math]x[/math] and [math]y[/math] such that [math]\overrightarrow{AO}=x\overrightarrow{AB}+y\overrightarrow{AC}[/math].

Let [math]ABC[/math] be a triangle satisfying [math]AB=3, BC=\sqrt{7}[/math], and [math]CA=2[/math]. Let [math]H[/math] be its orthocenter.

(a) [math]\overrightarrow{AH}\cdot\overrightarrow{AB}=[/math]
(A) [math]\overrightarrow{AB}\cdot\overrightarrow{AB}[/math]
(B) [math]\overrightarrow{AC}\cdot\overrightarrow{AC}[/math]
(C) [math]\overrightarrow{AB}\cdot\overrightarrow{BC}[/math]
(D) [math]\overrightarrow{BC}\cdot\overrightarrow{AC}[/math]
(E) [math]\overrightarrow{AB}\cdot\overrightarrow{AC}[/math]

(b) [math]\overrightarrow{AH}\cdot\overrightarrow{AC}=[/math]
(A) [math]\overrightarrow{AB}\cdot\overrightarrow{AC}[/math]
(B) [math]\overrightarrow{AB}\cdot\overrightarrow{AC}[/math]
(C) [math]\overrightarrow{AB}\cdot\overrightarrow{BC}[/math]
(D) [math]\overrightarrow{BC}\cdot\overrightarrow{AC}[/math]
(E) [math]\overrightarrow{AB}\cdot\overrightarrow{AC}[/math]

(c) Find constants [math]x[/math] and [math]y[/math] such that [math]\overrightarrow{AH}=x\overrightarrow{AB}+y\overrightarrow{AC}[/math].