In three-dimensional space, we use the three axes x, y, and z, and represent a point as (x, y, z). A vector is written with three components as →a = (a₁, a₂, a₃).
Magnitude and Operations
- |→a| = √(a₁² + a₂² + a₃²)
- Addition, subtraction, and scalar multiplication: calculate each component separately
Example ① For →a=(1,2,2) and →b=(3,−1,0), find |→a| and →a+→b.
Solution: |→a|=√(1+4+4)=3、→a+→b=(4,1,2)
Solution: |→a|=√(1+4+4)=3、→a+→b=(4,1,2)
Key Points
- Three-dimensional extension of plane vectors
- The magnitude is the square root of the sum of the squares of the three components
Practice Problems
- Magnitude of →a=(2,−1,2)
Answers and Explanations
- Answer: 3
√(4+1+4)=3。