Lesson 1: Vectors in R^n
Estimated time: 35-45 minutes
Learning Objectives
- Define vectors in R^n as ordered n-tuples
- Perform vector addition and scalar multiplication
- Compute the dot product and understand its properties
- Calculate vector length (norm) and understand unit vectors
- Determine when two vectors are orthogonal
Vectors in R^n
Vector in R^n: An ordered list of n real numbers, written as a column: v = (v_1, v_2, ..., v_n). R^n is the set of all such vectors. R^2 is the plane, R^3 is 3D space.
Vector Operations:
- Addition: u + v = (u_1+v_1, u_2+v_2, ..., u_n+v_n)
- Scalar multiplication: cv = (cv_1, cv_2, ..., cv_n)
Example
u = (1, -2, 3), v = (4, 0, -1).
u + v = (5, -2, 2). 3u = (3, -6, 9). u - 2v = (1-8, -2-0, 3+2) = (-7, -2, 5).
The Dot Product
Dot Product: u . v = u_1 v_1 + u_2 v_2 + ... + u_n v_n. The result is a scalar (a number), not a vector.
Example
u = (1, -2, 3), v = (4, 0, -1). u . v = 1(4) + (-2)(0) + 3(-1) = 4 + 0 - 3 = 1.
Properties of the Dot Product:
- u . v = v . u (commutative)
- u . (v + w) = u . v + u . w (distributive)
- (cu) . v = c(u . v)
- u . u ≥ 0, and u . u = 0 if and only if u = 0
Length and Unit Vectors
Length (Norm): ||v|| = sqrt(v . v) = sqrt(v_1^2 + v_2^2 + ... + v_n^2).
Unit Vector: A vector with length 1. To normalize v: v_hat = v / ||v||.
Example
v = (3, 4). ||v|| = sqrt(9 + 16) = sqrt(25) = 5. Unit vector: (3/5, 4/5).
Orthogonality
Orthogonal Vectors: Two vectors u and v are orthogonal if u . v = 0.
Example
u = (1, 2), v = (4, -2). u . v = 4 - 4 = 0. So u and v are orthogonal (perpendicular).
Check Your Understanding
1. Compute (2, -1, 3) . (1, 4, -2).
2. Find ||(-3, 4)||.
3. Are (1, 1, 1) and (1, -2, 1) orthogonal?
4. Normalize the vector (1, 2, 2).
Key Takeaways
- R^n = the set of all n-component vectors; operations are component-wise
- Dot product u . v produces a scalar; it is commutative and distributive
- Norm ||v|| = sqrt(v . v); unit vector = v / ||v||
- Vectors are orthogonal when their dot product is zero