linear-guest

90 Vectors in Space, n-Vectors
Example 49 of the dot product of two vectors from R 100 .
⎛
⎜
⎝
1
2
3
4
.
100
⎞ ⎛ ⎞
1
1
1
·
1
⎟ ⎜ ⎟
⎠ ⎝ . ⎠
1
= 1 + 2 + 3 + · · · + 100 = 1 .100.101 = 5050.
2
The sum above is the one Gauß, according to legend, could do in kindergarten.
Definition The length (or norm or magnitude) of an n-vector v is
‖v‖ := √ v v .
Example 50 of the norm of a vector from R 101 .
⎛
⎜
⎝
∥
1
2
3
4
.
101
⎞
∑
= √ 101
i 2 = √ 37, 961.
⎟
i=1
⎠
∥
Definition The angle θ between two vectors is determined by the formula
u v = ‖u‖‖v‖ cos θ .
Example 51 of an ⎛angle between ⎞ ⎛two ⎞vectors form R 101 .
1 1
2
0
3
1
( )
The angle between
4
and
0
is arccos 10,201
√ √ 37,916 51
.
⎜ ⎟ ⎜ ⎟
⎝ . ⎠ ⎝ . ⎠
101 1
Definition Two vectors are orthogonal (or perpendicular) if their dot
product is zero.
90