If A = <5 ,2 ,8 >, B = <6 ,5 ,3 > and C=A-B, what is the angle between A and C?

Aug 5, 2016

Depending upon the sense of measurement (clockwise or opposite),

the angle is ${59.45}^{o} \mathmr{and} {121.55}^{o}$

Explanation:

The angle between the vectors A and C is

$\theta = \text{arcsin} \left(| A \times C \frac{|}{| A | | B |}\right)$ or $\theta = \pi - \text{arcsin} \left(| A \times C \frac{|}{| A | | B |}\right)$

Here, $A = < 5 , 2 , 8 >$, and

$C = A - B = < 5 , 2 , 8 > - < 6 , 5 , 3 > = < - 1 , - 3 , 5 >$.

$A \times C = < 5 , 2 , 8 > \times < - 1 , - 3 , 5 >$

$< \left(2\right) \left(5\right) - \left(8\right) \left(- 3\right) , \left(8\right) \left(- 1\right) - \left(5\right) \left(5\right) , \left(5\right) \left(- 3\right) - \left(2\right) \left(- 1\right) >$

$= < 34 , - 33 , - 13 >$
So,

$| A | = \sqrt{{5}^{2} + {2}^{2} + {8}^{2}} = \sqrt{93}$

$| C | = \sqrt{{\left(- 1\right)}^{2} + {\left(- 3\right)}^{2} + {5}^{2}} = \sqrt{35}$

$| A \times C | = \sqrt{{\left(34\right)}^{2} + {\left(- 33\right)}^{2} + {\left(- 13\right)}^{2}} = \sqrt{2414}$.

The angle

$\theta = \text{arcsin} \sqrt{\frac{2414}{\left(93\right) \left(35\right)}} = {59.49}^{o}$

Depending upon the sense of measurement (clockwise or opposite),

the angle is ${59.49}^{o} \mathmr{and} {121.55}^{o}$

Aug 5, 2016

I got ${59.45}^{\circ}$ (using Cosine formula ...see A.S.Adikesavan's answer using Sine formula)

Explanation:

My understanding is that:

$\textcolor{b l u e}{\text{-------------------------------------------------------}}$
color(blue)("| ")"Given vectors: "
color(blue)("| ")color(white)("XXX")color(blue)( vecu=< u_1,u_2,u_3 >" and"color(white)("XXXXX")color(blue)("|"
$\textcolor{b l u e}{\text{| ")color(white)("XXX")color(blue)(vecv = < v_1,v_2,v_3>color(white)("XXXXXxXXX")color(blue)("|}}$

$\textcolor{b l u e}{\text{| }}$For an angle $\textcolor{b l u e}{\theta}$ between $\textcolor{b l u e}{\vec{u}}$ and $\textcolor{b l u e}{\vec{v}}$$\textcolor{w h i t e}{\text{XXXXxX")color(blue)("|}}$
$\textcolor{b l u e}{\text{| ") color(white)("XXX")color(blue)(sin(theta)=(vecu * vecv)/(abs(vecu)*abs(vecv))color(white)("XXXXXXXxXX")color(blue)("|}}$

$\textcolor{b l u e}{\text{| }}$where$\textcolor{w h i t e}{\text{XXXXXXXXXXXX}}$
$\textcolor{b l u e}{\text{| ")color(white)("XXX")color(blue)(vecu * vecv = u_1 * v_1+u_2 * v_2 + u_3 * v_3)color(white)("X")color(blue)("|}}$
$\textcolor{b l u e}{\text{| ")color(white)("XXX")color(blue)(abs(vecu)=sqrt(u_1^2+u_2^2+u_3^2))color(white)("XXXXXXxX")color(blue)("|}}$
$\textcolor{b l u e}{\text{| ")color(white)("XXX")color(blue)(abs(vecv)=sqrt(v_1^2+v_2^2+v_3^2))color(white)("XXXXXXXX")color(blue)("|}}$
$\textcolor{b l u e}{\text{-------------------------------------------------------}}$

In this case, we are given
$\textcolor{w h i t e}{\text{XXX}} \textcolor{red}{\vec{A} = < 5 , 2 , 8 >} , \textcolor{red}{\vec{B} = < 6 , 5 , 3 >} , \mathmr{and} \textcolor{red}{\vec{C} = \vec{A} - \vec{B}}$
$\textcolor{w h i t e}{\text{XXX}} \rightarrow \textcolor{red}{\vec{C} = < - 1 , - 3 , 5 >}$

$\textcolor{red}{\vec{A}} \cdot \textcolor{red}{\vec{C}} = \left(5\right) \left(- 1\right) + \left(2\right) \left(- 3\right) + \left(8\right) \left(5\right) \textcolor{red}{= 29}$

$\textcolor{red}{\left\mid \vec{A} \right\mid} = \sqrt{{5}^{2} + {2}^{2} + {8}^{2}} \textcolor{red}{= \sqrt{93}}$
$\textcolor{red}{\left\mid \vec{C} \right\mid} = \sqrt{{1}^{1} + {3}^{2} + {5}^{2}} \textcolor{red}{= \sqrt{35}}$

Therefore
$\textcolor{w h i t e}{\text{XXX}} \textcolor{g r e e n}{\cos \left(\theta\right)} = \frac{\textcolor{red}{29}}{\textcolor{red}{\sqrt{93} \cdot \sqrt{35}}}$

Using a calculator:
$\textcolor{w h i t e}{\text{XXX}} \textcolor{g r e e n}{\cos \left(\theta\right)} = 0.508303$
and
color(white)("XXX")color(green)(theta) = "arccos"(0.508303) = color(green)(59.45^@)