En esta primera parte de los ejercicios de multiplicación de vectores se presentan ejercicios en donde los vectores están dados con el punto inicial y el punto final. Los ejercicios se resuelven de la siguiente manera:
Tenemos los vectores:
\begin{align}{UPI} = [ x_{1} ,y_{1,} z_{1} ] \end{align}
\begin{align}{UPF} = [ x_{2} ,y_{2,}z_{2} ]\end{align}
\begin{align}{VPI} = [ a_{1} ,b_{1,} c_{1} ]\end{align}
\begin{align}{VPF} = [ a_{2} ,b_{2,} c_{2} ]\end{align}
*** UPI = punto inicial del vector U, UPF= punto final del vector U
\begin{align}U = [ x_{2} -x_{1} ,y_{2} -y_{1,} z_{2} -z_{1} ] = [ u_{1} ,u_{2,} u_{3} ]\end{align}
\begin{align}V = [ a_{2} -a_{1} ,b_{2} -b_{1,} c_{2} -c_{1} ] = [ v_{1} ,v_{2,} v_{3} ]\end{align}
\begin{align}\overset{}{} \overset{\rightarrow}{u} \times \overset{\rightarrow}{v} = (
u_{2} v_{3} -v_{2} u_{3} ) i- ( u_{1} v_{3} -v_{1} u_{3} ) j+ ( u_{1} v_{2}
-v_{1} u_{2} ) k\end{align}
Ejercicios de multiplicación de vectores
1.- \begin{align}{UPI} = [ 3, -2, 4 ] \end{align}
\begin{align}{UPF} = [ 6, -2, 6 ]\end{align}
\begin{align}{VPI} = [ 3, 4, 1 ]\end{align}
\begin{align}{VPF} = [ -2, 4, 5 ]\end{align}
*** UPI = punto inicial del vector U, UPF= punto final del vector U
\begin{align}U = [6-3 ,-2- (-2), 6-4 ] = [3 , 0, 2 ]\end{align}
\begin{align}V = [ -2-3 , 4-4, 5-1 ] = [ -5, 0, 4]\end{align}
\begin{align}\overset{}{} \overset{\rightarrow}{u} \times \overset{\rightarrow}{v} = (
u_{2} v_{3} -v_{2} u_{3} ) i- ( u_{1} v_{3} -v_{1} u_{3} ) j+ ( u_{1} v_{2}
-v_{1} u_{2} ) k\end{align}
\begin{align}\overset{}{} \overset{\rightarrow}{u} \times \overset{\rightarrow}{v} = ( 0
(4) – (0) 2 ) i- (3 (4 ) -(-5) 2 ) j+ ( 3 ( 0 ) -(-5) ( 0) ) k\end{align}
\begin{align}\overset{}{} \overset{\rightarrow}{u} \times \overset{\rightarrow}{v} = (
0) i- (12+10) j+ ( 0) k\end{align}
\begin{align}\overset{}{} \overset{\rightarrow}{u} \times \overset{\rightarrow}{v} =
0i-22j+0k\end{align}
2.- \begin{align}{UPI} = [1, 3, 5 ] \end{align}
\begin{align}{UPF} = [ 6, 5, 4 ]\end{align}
\begin{align}{VPI} = [3, 9, 8 ]\end{align}
\begin{align}{VPF} = [ 7, 5, 6 ]\end{align}
*** UPI = punto inicial del vector U, UPF= punto final del vector U
\begin{align}U = [6-1, 5-3, 4-5 ] = [ 5, 2, -1]\end{align}
\begin{align}V = [7-3, 5-9, 6-8 ] = [4, -4, -2 ]\end{align}
\begin{align}\overset{}{} \overset{\rightarrow}{u} \times \overset{\rightarrow}{v} = (
u_{2} v_{3} -v_{2} u_{3} ) i- ( u_{1} v_{3} -v_{1} u_{3} ) j+ ( u_{1} v_{2}
-v_{1} u_{2} ) k\end{align}
\begin{align}\overset{}{} \overset{\rightarrow}{u} \times \overset{\rightarrow}{v} = ( 2
( -2) – (-4) (-1) ) i- (5 ( -2 ) -(4) (-1) ) j+ ( 5 ( -4) -(4) ( 2) ) k\end{align}
\begin{align}\overset{}{} \overset{\rightarrow}{u} \times \overset{\rightarrow}{v} = (
-4-4) i- (-10+4) j+ ( -20-8 ) k\end{align}
\begin{align}\overset{}{} \overset{\rightarrow}{u} \times \overset{\rightarrow}{v} =
-8i+6j-28k\end{align}
3.- \begin{align}{UPI} = [ 1, 9, 8 ] \end{align}
\begin{align}{UPF} = [6, 5, 3 ]\end{align}
\begin{align}{VPI} = [ -2, -1, -2 ]\end{align}
\begin{align}{VPF} = [ -3, -1, -2 ]\end{align}
*** UPI = punto inicial del vector U, UPF= punto final del vector U
\begin{align}U = [6-1, 5-9, 3-8 ] = [ 5, -4, -5 ]\end{align}
\begin{align}V = [-3-(-2), -1-(-1), -2-(-2) ] = [ -1, 0, 0 ]\end{align}
\begin{align}\overset{}{} \overset{\rightarrow}{u} \times \overset{\rightarrow}{v} = (
u_{2} v_{3} -v_{2} u_{3} ) i- ( u_{1} v_{3} -v_{1} u_{3} ) j+ ( u_{1} v_{2}
-v_{1} u_{2} ) k\end{align}
\begin{align}\overset{}{} \overset{\rightarrow}{u} \times \overset{\rightarrow}{v} = ( -4
( 0) – (0) -5 ) i- (5 ( 0 ) -(-1) (-5)) j+ ( 5 ( 0 ) -(-1) ( -4) ) k\end{align}
\begin{align}\overset{}{} \overset{\rightarrow}{u} \times \overset{\rightarrow}{v} = (
0) i- (-5) j+ ( -4 ) k\end{align}
\begin{align}\overset{}{} \overset{\rightarrow}{u} \times \overset{\rightarrow}{v} =
0i-5j-4k\end{align}
4.- \begin{align}{UPI} = [6, 5, -2 ] \end{align}
\begin{align}{UPF} = [-3, 1, 5 ]\end{align}
\begin{align}{VPI} = [ 6, 15, 8 ]\end{align}
\begin{align}{VPF} = [ -6, 5, -4 ]\end{align}
*** UPI = punto inicial del vector U, UPF= punto final del vector U
\begin{align}U = [-3-6, 1-5, 5-(-2) ] = [-9, -4, 7 ]\end{align}
\begin{align}V = [-6-6, 5-15, -4-8 ] = [-12, -10, -12 ]\end{align}
\begin{align}\overset{}{} \overset{\rightarrow}{u} \times \overset{\rightarrow}{v} = (
u_{2} v_{3} -v_{2} u_{3} ) i- ( u_{1} v_{3} -v_{1} u_{3} ) j+ ( u_{1} v_{2}
-v_{1} u_{2} ) k\end{align}
\begin{align}\overset{}{} \overset{\rightarrow}{u} \times \overset{\rightarrow}{v} = ( -4
( -12) – (-10) 7 ) i- (-9 ( -12 ) -(-12) 7 ) j+ ( -9 ( -10 ) -(-12) ( -4) ) k\end{align}
\begin{align}\overset{}{} \overset{\rightarrow}{u} \times \overset{\rightarrow}{v} = (
48+70) i- (108+84) j+ ( 90-48 ) k\end{align}
\begin{align}\overset{}{} \overset{\rightarrow}{u} \times \overset{\rightarrow}{v} =
118i-192j+42k\end{align}
5.- \begin{align}{UPI} = [ 1, 2, 3 ] \end{align}
\begin{align}{UPF} = [ 3, 2, 1 ]\end{align}
\begin{align}{VPI} = [ 1, 2, 3 ]\end{align}
\begin{align}{VPF} = [ 3, 2, 1 ]\end{align}
*** UPI = punto inicial del vector U, UPF= punto final del vector U
\begin{align}U = [3-1, 2-2, 1-3 ] = [2, 0, -2]\end{align}
\begin{align}V = [3-1, 2-2, 1-3 ] = [2,0,-2]\end{align}
\begin{align}\overset{}{} \overset{\rightarrow}{u} \times \overset{\rightarrow}{v} = (
u_{2} v_{3} -v_{2} u_{3} ) i- ( u_{1} v_{3} -v_{1} u_{3} ) j+ ( u_{1} v_{2}
-v_{1} u_{2} ) k\end{align}
\begin{align}\overset{}{} \overset{\rightarrow}{u} \times \overset{\rightarrow}{v} = ( 0
( -2) – 0(-2) ) i- (2 ( -2 ) -2(-2)) j+ ( 2 ( 0 ) -(2) ( 0) ) k\end{align}
\begin{align}\overset{}{} \overset{\rightarrow}{u} \times \overset{\rightarrow}{v} = (
0) i- (-4+4) j+ ( 0 ) k\end{align}
\begin{align}\overset{}{} \overset{\rightarrow}{u} \times \overset{\rightarrow}{v} =
0i+0j+0k\end{align}
6.- \begin{align}{UPI} = [6, 5, 7] \end{align}
\begin{align}{UPF} = [12, 15, 13 ]\end{align}
\begin{align}{VPI} = [1, 5, 3 ]\end{align}
\begin{align}{VPF} = [3, 9, 5 ]\end{align}
*** UPI = punto inicial del vector U, UPF= punto final del vector U
\begin{align}U = [12-6, 15-5, 13-7 ] = [6, 10, 6 ]\end{align}
\begin{align}V = [3-1, 9-5, 5-3] = [2, 4, 2 ]\end{align}
\begin{align}\overset{}{} \overset{\rightarrow}{u} \times \overset{\rightarrow}{v} = (
u_{2} v_{3} -v_{2} u_{3} ) i- ( u_{1} v_{3} -v_{1} u_{3} ) j+ ( u_{1} v_{2}
-v_{1} u_{2} ) k\end{align}
\begin{align}\overset{}{} \overset{\rightarrow}{u} \times \overset{\rightarrow}{v} = ( 10
( 2) – (4) 6 ) i- ( 6( 2) -(2) 6 ) j+ ( 6 ( 4 ) -(2) ( 10) ) k\end{align}
\begin{align}\overset{}{} \overset{\rightarrow}{u} \times \overset{\rightarrow}{v} = (
20-24) i- (12-12) j+ (24-20 ) k\end{align}
\begin{align}\overset{}{} \overset{\rightarrow}{u} \times \overset{\rightarrow}{v} =
-4i-0j+4k\end{align}
7.- \begin{align}{UPI} = [-3, -5, -6 ] \end{align}
\begin{align}{UPF} = [1, 5, 4 ]\end{align}
\begin{align}{VPI} = [5, 6, 7 ]\end{align}
\begin{align}{VPF} = [-1, -2, -4 ]\end{align}
*** UPI = punto inicial del vector U, UPF= punto final del vector U
\begin{align}U = [1+3, 5+5, 4+6 ] = [4, 10, 10 ]\end{align}
\begin{align}V = [-1-5, -2-6, -4-7 ] = [-6, -8, -11 ]\end{align}
\begin{align}\overset{}{} \overset{\rightarrow}{u} \times \overset{\rightarrow}{v} = (
u_{2} v_{3} -v_{2} u_{3} ) i- ( u_{1} v_{3} -v_{1} u_{3} ) j+ ( u_{1} v_{2}
-v_{1} u_{2} ) k\end{align}
\begin{align}\overset{}{} \overset{\rightarrow}{u} \times \overset{\rightarrow}{v} = ( 10
( -11) – (-8) 10 ) i- (4 ( -11 ) -(-6) 10 ) j+ ( 4 ( -8 ) -(-6) ( 10) ) k\end{align}
\begin{align}\overset{}{} \overset{\rightarrow}{u} \times \overset{\rightarrow}{v} = (
-110+80) i- (-44+60) j+ ( -32+60 ) k\end{align}
\begin{align}\overset{}{} \overset{\rightarrow}{u} \times \overset{\rightarrow}{v} =
-30i-16j+28k\end{align}
En la parte 2 de este artículo se verán ejercicios en donde los vectores se dan con sus componentes en “x”, “y” y “z”.
