Step by step
The question
Question: find the unit vector in the direction of (2, 3, 6).
v = [2; 3; 6]
A unit vector has length 1.
To get one in the same direction, we divide v by its length.
Dividing every entry by a number makes the vector that many times shorter, so the length becomes 1.
unit vector: length 1, same direction
u = v ÷ ‖v‖
1 Find the length
The length: every entry squared, added, then the square root.
‖v‖ = √22 + 32 + 62
‖v‖ = √4 + 9 + 36
‖v‖ = √49
49 is 7 squared.
So the length is 7.
‖v‖ = 7
2 Divide every entry
Every entry divided by 7.
u = [2 over 7; 3 over 7; 6 over 7]
Careful!
Every entry is divided by the length, not only the first one.
Careful: divide EVERY entry by the length
3 Check
Let's check: the length of u must be 1.
Its entries squared add up to 1, and the root of 1 is 1.
So the answer is right!
(27)2 + (37)2 + (67)2 = 449 + 949 + 3649 = 4949
= 1
Let's look back.
First: find the length.
Then: divide every entry.
Last: check.
Answer
Careful: divide EVERY entry by the length