# Lecture 7 Euclidean Spaces (Completed)

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Euclidean Spaces Def: Let β be the set of all real numbers. We regard β geometrically as the Euclidean line . Ex: Plot β2, 1.5, and π on the Euclidean line. Def: Each ordered pair (π₯, π¦) of real numbers represents a point in the π₯π¦ βplane. The set of all such ordered pairs is Euclidean 2-space, which is denoted β ΰ¬Ά , and often called the plane . The axes meet at the point (0,0) which is called the origin . Note: alternatively, we may use π₯ ΰ¬΅ for the horizontal axis, π₯ ΰ¬Ά for the vertical axis, and call β ΰ¬Ά the π₯ ΰ¬΅ π₯ ΰ¬Ά -plane. Vectors in β ΰ¬Ά Def: A pair of real numbers π and π produces a two dimensional vector π£ β. We write π£ β as a column and visualize π£ β as an arrow in β ΰ¬Ά with tail located at the origin and head at the point (π, π). Lecture 7: Euclidean Spaces Math 220 Fall 2023 Lecture Notes Page 1
π is called the first component of π£ β and π is the second component . Ex: Sketch π£ β = ΰ΅€ β1 3 ΰ΅¨ and π€ α¬α¬β = α 2 β4 α in β ΰ¬Ά . Def: The zero vector 0 α¬β has components π = 0 and π = 0. Def: Two vectors are equal if their corresponding entries are equal. Ex: Determine if each pair of vectors are equal vectors. (1) α 4 7 α and α 7 4 α (2) ΰ΅€ 1 5 ΰ΅¨ and ΰ΅€ 4 β 3 4 + 1 ΰ΅¨ Math 220 Fall 2023 Lecture Notes Page 2
Vector Operations Vector addition: we add two vectors by adding component-wise. If π£ β = ΰ΅€ π£ ΰ¬΅ π£ ΰ¬Ά ΰ΅¨ and π€ α¬α¬β = α π€ ΰ¬΅ π€ ΰ¬Ά α , π£ β + π€ α¬α¬β = Geometrically, we add vectors using the Parallelogram Law : v α¬β + π€ α¬α¬β lies along the diagonal of the parallelogram with sides π£ β and π€ α¬α¬β. Ex: Let π£ β = α 3 1 α and π€ α¬α¬β = α β1 2 α. Find π£ β + π€ α¬α¬β geometrically using the Parallelogram Law. Then confirm your answer algebraically. Scalar multiplication : let π be a scalar (real number) and π£ β be a vector. The scalar multiple ππ£ β is the vector with components ππ£ ΰ¬΅ and ππ£ ΰ¬Ά . Ex: Let π£ β = α 3 1 α . Find 2π£ β, ΰ¬΅ ΰ¬Ά β―π£ β, and βπ£ β. How does multiplication by each scalar change the vector geometrically? Math 220 Fall 2023 Lecture Notes Page 3
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