Lecture 7 Euclidean Spaces (Completed)

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Euclidean Spaces Def: Let ℝ be the set of all real numbers. We regard ℝ geometrically as the Euclidean line . Ex: Plot −2, 1.5, and 𝜋 on the Euclidean line. Def: Each ordered pair (𝑥, 𝑦) of real numbers represents a point in the 𝑥𝑦 −plane. The set of all such ordered pairs is Euclidean 2-space, which is denoted ℝ , and often called the plane . The axes meet at the point (0,0) which is called the origin . Note: alternatively, we may use 𝑥 for the horizontal axis, 𝑥 for the vertical axis, and call ℝ the 𝑥 𝑥 -plane. Vectors in ℝ Def: A pair of real numbers 𝑎 and 𝑏 produces a two dimensional vector 𝑣 ⃑. We write 𝑣 ⃑ as a column and visualize 𝑣 ⃑ as an arrow in ℝ with tail located at the origin and head at the point (𝑎, 𝑏). Lecture 7: Euclidean Spaces Math 220 Fall 2023 Lecture Notes Page 1
𝑎 is called the first component of 𝑣 and 𝑏 is the second component . Ex: Sketch 𝑣 ⃗ = ൤ −1 3 ൨ and 𝑤 ሬሬ⃗ = ቂ 2 −4 ቃ in ℝ . Def: The zero vector 0 ሬ⃑ has components 𝑎 = 0 and 𝑏 = 0. Def: Two vectors are equal if their corresponding entries are equal. Ex: Determine if each pair of vectors are equal vectors. (1) ቂ 4 7 ቃ and ቂ 7 4 (2) ൤ 1 5 ൨ and ൤ 4 − 3 4 + 1 Math 220 Fall 2023 Lecture Notes Page 2
Vector Operations Vector addition: we add two vectors by adding component-wise. If 𝑣 ⃗ = ൤ 𝑣 𝑣 and 𝑤 ሬሬ⃗ = ቂ 𝑤 𝑤 ቃ , 𝑣 ⃗ + 𝑤 ሬሬ⃗ = Geometrically, we add vectors using the Parallelogram Law : v ሬ⃗ + 𝑤 ሬሬ⃗ lies along the diagonal of the parallelogram with sides 𝑣 and 𝑤 ሬሬ⃗. Ex: Let 𝑣 ⃗ = ቂ 3 1 ቃ and 𝑤 ሬሬ⃗ = ቂ −1 2 ቃ. Find 𝑣 ⃗ + 𝑤 ሬሬ⃗ geometrically using the Parallelogram Law. Then confirm your answer algebraically. Scalar multiplication : let 𝑐 be a scalar (real number) and 𝑣 ⃗ be a vector. The scalar multiple 𝑐𝑣 ⃗ is the vector with components 𝑐𝑣 and 𝑐𝑣 . Ex: Let 𝑣 ⃗ = ቂ 3 1 ቃ . Find 2𝑣 ⃗, ⎯𝑣 ⃗, and −𝑣 ⃗. How does multiplication by each scalar change the vector geometrically? Math 220 Fall 2023 Lecture Notes Page 3
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