This document contains solutions to vector algebra problems. It provides the steps to calculate: (1) the magnitude of given vectors, (2) the sum and difference of vectors, and (3) scalar multiplication of vectors. For each problem, it lists the vectors, shows the calculation steps, and states the final result. The purpose is to demonstrate how to perform basic vector operations through worked examples.

1. Matemáticas Académicas 4ºESO
© Marta Martín Sierra 1
ACTIVIDADES:
01. Dados dos vectores u
r
= (1, 3) y v
r
= (2, – 2), calcula:
(a) |u
r
| (b) |v
r
| (c) u
r
+ v
r
(d) u
r
– v
r
(e) 3u
r
(f) 5u
r
– 3v
r
RESOLUCIÓN:
(a) |u
r
|
|u
r
| = 22
31 +
|u
r
| = 10 unidades
(b) |v
r
|
|v
r
| = 22
22 )(−+
|v
r
| = 8 u
(c) u
r
+ v
r
u
r
= (1, 3) y v
r
= (2, – 2),
u
r
+ v
r
=
= (1 + 2, 3 + (– 2)) =
= (3, 1)
(d) u
r
– v
r
u
r
= (1, 3) y v
r
= (2, – 2),
u
r
– v
r
= (1, 3) – (2, – 2) =
= (1 – 2, 3 – (– 2)) =
= (– 1, 5)
(e) 3u
r
u
r
= (1, 3)
3· u
r
=
= 3·(1, 3) =
= (3, 9)
(f) 5u
r
– 3v
r
u
r
= (1, 3) y v
r
= (2, – 2)
5· (1, 3) – 3·(2, – 2) =
Recomendación: Multiplicamos el segundo vector por 3 para transformar la
operación en una resta sencilla de vectores en el paso siguiente:
= (5, 15) – (6, – 6) =
= (– 1, 21)
02. Dados los vectores u
r
= (– 2, 3) y v
r
= (6, – 1), calcula:
(a) |u
r
| (b) |v
r
| (c) u
r
+ v
r
(d) |u
r
+ v
r
|
(e) u
r
– v
r
(f) 4u
r
(g) 5u
r
– 6v
r
(h) 2v
r
+ 3u
r
RESOLUCIÓN:
(a) |u
r
|
|u
r
| = 22
3)2( +−
|u
r
| = 13 unidades
(b) |v
r
|
|v
r
| = 22
)1(6 −+

2. Geometría Analítica
TEMA 09 – GEOMETRÍA ANALÍTICA EN EL PLANO – RESUELTOS2
|v
r
| = 37 unidades
(c) u
r
+ v
r
u
r
+ v
r
= (– 2, 3) + (6, – 1)
= (– 2 + 6, 3 + (– 1)) =
= (4, 2)
(d) |u
r
+ v
r
|
|u
r
+ v
r
| =
Como ya tenemos hecho u
r
+ v
r
= (4, 2)
|u
r
+ v
r
| = 22
24 +
|u
r
+ v
r
| = 20 unidades
(e) u
r
– v
r
u
r
– v
r
= (– 2, 3) – (6, – 1)=
= (– 2 – 6, 3 – (– 1)) =
= (– 8, 4)
(f) 4u
r
u
r
= (– 2, 3)
4· u
r
=
= 4· (– 2, 3) =
= (– 8, 12)
(g) 5u
r
– 6v
r
5u
r
– 6v
r
= 5·(– 2, 3) – 6·(6, – 1) =
Recomendación: Multiplicamos el segundo vector por 3 para transformar la
operación en una resta sencilla de vectores en el paso siguiente:
= (–10, 15) – (36, – 6) =
= (– 10 – 36, 15 – (–6)) =
= (– 46, 21) =
(h) 2v
r
+ 3u
r
2v
r
+ 3u
r
= 2·(6, – 1) + 3·(– 2, 3) =
= (12, – 2) + (– 6, 9) =
= (6, 7)