3.16. Контрольные задания
Даны векторы
и
.
Найти скалярное произведение
.При каком значении k векторы
и
ортогональны?
№ вар |
|
|
||||
a1 |
a2 |
a3 |
b1 |
b2 |
b3 |
|
1 |
1 |
2 |
0 |
0 |
-1 |
2 |
2 |
1 |
2 |
–1 |
0 |
-1 |
1 |
3 |
–2 |
2 |
4 |
1 |
1 |
0 |
4 |
1 |
0 |
1 |
0 |
2 |
1 |
5 |
2 |
1 |
0 |
1 |
0 |
1 |
6 |
0 |
1 |
–1 |
2 |
1 |
-1 |
7 |
2 |
–1 |
0 |
0 |
-1 |
-1 |
8 |
0 |
1 |
1 |
1 |
1 |
0 |
9 |
1 |
0 |
1 |
3 |
2 |
1 |
10 |
0 |
2 |
–1 |
-2 |
2 |
-1 |
11 |
1 |
0 |
–1 |
-1 |
3 |
-1 |
12 |
–3 |
–1 |
0 |
0 |
1 |
-1 |
13 |
0 |
–2 |
2 |
3 |
1 |
2 |
14 |
1 |
1 |
1 |
0 |
1 |
2 |
15 |
0 |
–1 |
–2 |
1 |
-1 |
2 |
16 |
1 |
2 |
0 |
-1 |
-1 |
2 |
17 |
1 |
–2 |
–1 |
0 |
-1 |
1 |
18 |
0 |
–2 |
1 |
1 |
1 |
0 |
19 |
0 |
2 |
1 |
-1 |
-2 |
2 |
20 |
2 |
1 |
0 |
-1 |
0 |
1 |
21 |
0 |
1 |
–1 |
2 |
-1 |
-1 |
22 |
2 |
–1 |
0 |
0 |
-1 |
-1 |
23 |
0 |
1 |
1 |
1 |
1 |
0 |
24 |
–2 |
0 |
1 |
3 |
2 |
1 |
25 |
0 |
1 |
–2 |
-1 |
3 |
-1 |
26 |
1 |
–1 |
–1 |
2 |
3 |
-1 |
27 |
2 |
1 |
1 |
0 |
1 |
-1 |
28 |
0 |
2 |
2 |
3 |
1 |
-1 |
29 |
1 |
1 |
1 |
0 |
1 |
2 |
30 |
0 |
–1 |
–2 |
-1 |
1 |
-2 |
Выяснить, является ли система векторов
,
,
линейно зависимой или линейно независимой;
определить ранг системы векторов.
№ вар |
|
|
|
||||||
a1 |
a2 |
a3 |
b1 |
b2 |
b3 |
с1 |
с2 |
с3 |
|
|
|
2 |
3 |
1 |
3 |
4 |
1 |
4 |
5 |
1 |
|
|
7 |
6 |
5 |
1 |
2 |
3 |
3 |
2 |
1 |
|
|
1 |
–2 |
2 |
1 |
3 |
1 |
2 |
1 |
3 |
|
|
3 |
3 |
1 |
1 |
0 |
1 |
4 |
3 |
2 |
|
|
2 |
2 |
3 |
3 |
4 |
5 |
1 |
2 |
2 |
|
|
3 |
1 |
–4 |
1 |
2 |
5 |
4 |
3 |
1 |
|
|
4 |
2 |
3 |
5 |
4 |
7 |
1 |
2 |
4 |
|
|
3 |
3 |
3 |
4 |
6 |
8 |
1 |
3 |
5 |
|
|
9 |
5 |
7 |
4 |
4 |
5 |
5 |
1 |
2 |
|
|
4 |
6 |
8 |
1 |
5 |
3 |
3 |
1 |
5 |
|
|
2 |
3 |
1 |
3 |
4 |
1 |
8 |
11 |
3 |
|
|
4 |
4 |
4 |
1 |
2 |
3 |
3 |
2 |
1 |
|
|
3 |
–1 |
5 |
1 |
3 |
1 |
2 |
1 |
3 |
|
|
–3 |
–3 |
–1 |
1 |
0 |
1 |
4 |
3 |
2 |
|
|
–2 |
–2 |
–3 |
3 |
4 |
5 |
1 |
2 |
2 |
|
|
7 |
4 |
–3 |
1 |
2 |
5 |
4 |
3 |
1 |
|
|
–4 |
–2 |
–3 |
5 |
4 |
7 |
1 |
2 |
4 |
|
|
7 |
9 |
11 |
4 |
6 |
8 |
1 |
3 |
5 |
|
|
–1 |
3 |
3 |
4 |
4 |
5 |
5 |
1 |
2 |
|
|
–2 |
4 |
–2 |
1 |
5 |
3 |
3 |
1 |
5 |
|
|
2 |
3 |
1 |
3 |
4 |
1 |
–4 |
–5 |
–1 |
|
|
5 |
2 |
–1 |
1 |
2 |
3 |
3 |
2 |
1 |
|
|
5 |
0 |
8 |
1 |
3 |
1 |
2 |
1 |
3 |
|
|
7 |
6 |
3 |
1 |
0 |
1 |
4 |
3 |
2 |
|
|
5 |
6 |
8 |
3 |
4 |
5 |
1 |
2 |
2 |
|
|
–3 |
–1 |
4 |
1 |
2 |
5 |
4 |
3 |
1 |
|
|
9 |
10 |
6 |
5 |
4 |
7 |
1 |
2 |
4 |
|
|
5 |
9 |
13 |
4 |
6 |
8 |
1 |
3 |
5 |
|
|
1 |
–3 |
–3 |
4 |
4 |
5 |
5 |
1 |
2 |
|
|
–1 |
9 |
1 |
1 |
5 |
3 |
3 |
1 |
5 |
Разложить вектор по базису ( , ) и сделать проверку.
№ вар |
|
|
|
|||||||||
a1 |
a2 |
a3 |
a4 |
b1 |
b2 |
b3 |
b4 |
c1 |
c2 |
c3 |
c4 |
|
|
|
14 |
–2 |
0 |
12 |
2 |
–4 |
5 |
3 |
12 |
2 |
–5 |
9 |
|
|
10 |
1 |
13 |
24 |
7 |
0 |
9 |
16 |
3 |
1 |
4 |
8 |
|
|
7 |
13 |
–3 |
17 |
5 |
2 |
7 |
14 |
2 |
11 |
–10 |
3 |
|
|
3 |
2 |
3 |
2 |
4 |
1 |
3 |
8 |
7 |
–1 |
0 |
6 |
|
|
–2 |
–6 |
–1 |
5 |
6 |
12 |
–7 |
11 |
2 |
3 |
3 |
8 |
|
|
14 |
14 |
–6 |
22 |
9 |
11 |
–1 |
19 |
5 |
3 |
–5 |
3 |
|
|
–1 |
11 |
–7 |
3 |
2 |
4 |
1 |
7 |
3 |
–7 |
8 |
4 |
|
|
6 |
3 |
2 |
11 |
1 |
6 |
–7 |
0 |
5 |
–3 |
9 |
11 |
|
|
–1 |
4 |
2 |
5 |
1 |
3 |
0 |
4 |
2 |
–1 |
–2 |
–1 |
|
|
3 |
11 |
–12 |
2 |
1 |
2 |
–5 |
–2 |
2 |
9 |
–7 |
4 |
|
|
16 |
–6 |
5 |
–3 |
2 |
–4 |
5 |
3 |
12 |
2 |
–5 |
9 |
|
|
4 |
–1 |
5 |
8 |
7 |
0 |
9 |
16 |
3 |
1 |
4 |
8 |
|
|
–3 |
9 |
–17 |
–11 |
5 |
2 |
7 |
14 |
2 |
11 |
–10 |
3 |
|
|
11 |
0 |
3 |
14 |
4 |
1 |
3 |
8 |
7 |
–1 |
0 |
6 |
|
|
4 |
9 |
–11 |
3 |
6 |
12 |
–7 |
11 |
2 |
3 |
3 |
8 |
|
|
–1 |
5 |
9 |
13 |
9 |
11 |
–1 |
19 |
5 |
3 |
–5 |
3 |
|
|
6 |
–3 |
9 |
11 |
2 |
4 |
1 |
7 |
3 |
–7 |
8 |
4 |
|
|
–3 |
15 |
–25 |
–11 |
1 |
6 |
–7 |
0 |
5 |
–3 |
9 |
11 |
|
|
3 |
2 |
–2 |
3 |
1 |
3 |
0 |
4 |
2 |
–1 |
–2 |
–1 |
|
|
–1 |
–7 |
2 |
–6 |
1 |
2 |
–5 |
–2 |
2 |
9 |
–7 |
4 |
|
|
–10 |
–6 |
10 |
–6 |
2 |
–4 |
5 |
3 |
12 |
2 |
–5 |
9 |
|
|
–4 |
1 |
–5 |
–8 |
7 |
0 |
9 |
16 |
3 |
1 |
4 |
8 |
|
|
3 |
–9 |
17 |
11 |
5 |
2 |
7 |
14 |
2 |
11 |
–10 |
3 |
|
|
1 |
3 |
6 |
10 |
4 |
1 |
3 |
8 |
7 |
–1 |
0 |
6 |
|
|
8 |
15 |
–4 |
19 |
6 |
12 |
–7 |
11 |
2 |
3 |
3 |
8 |
|
|
4 |
8 |
4 |
16 |
9 |
11 |
–1 |
19 |
5 |
3 |
–5 |
3 |
|
|
1 |
1 |
–6 |
10 |
2 |
4 |
1 |
7 |
3 |
–7 |
8 |
4 |
|
|
–4 |
9 |
–16 |
–11 |
1 |
6 |
–7 |
0 |
5 |
–3 |
9 |
11 |
|
|
0 |
7 |
2 |
9 |
1 |
3 |
0 |
4 |
2 |
–1 |
–2 |
–1 |
|
|
0 |
–5 |
–3 |
–8 |
1 |
2 |
–5 |
–2 |
2 |
9 |
–7 |
4 |