Vectors：練習問題

Two vectors are defined as \( u = [3, 4] \) and \( v = [-1, 2] \). Calculate the dot product (scalar product) of these two vectors.

Given two 3D vectors \( u = [1, 0, -2] \) and \( v = [3, 2, 1] \), calculate the dot product \( u \cdot v \).

Consider two vectors \( u = [2, 6] \) and \( v = [10, 2] \). A convex combination of these vectors is defined as \( w = \alpha u + \beta v \). If \( \alpha = 0.4 \), what is the resulting vector \( w \)?

In the context of vectors, which of the following best describes a convex combination of two vectors \( u \) and \( v \)?

A vector \( a \) is represented as an arrow with its tail at the origin \( (0, 0) \) and its head at \( (4, -3) \). A second vector \( b \) is defined as \( [1, 5] \). If a scalar multiplication \( 2a \) is performed followed by vector addition \( 2a + b \), what are the coordinates of the resulting vector's head?

Consider the 2D vector \(v = [3, 4]\). Calculate the result of the scalar-vector multiplication \(2v\).

Calculate the dot product (scalar product) of the vectors \(u = [1, 2, 3]\) and \(v = [4, 5, 6]\).

Define a convex combination of two 3D vectors \( u \) and \( v \) using scalars \( \alpha \) and \( \beta \), and state the necessary constraints on these scalars.

Consider two vectors in a 2D plane: \( u = [4, -1] \) and \( v = [2, 3] \).

a. Calculate the vector addition \( u + v \). (1 point)
b. Calculate the scalar multiplication \( 3u \). (1 point)
c. Calculate the dot product \( u \cdot v \). (1 point)

In the study of Vectors, a convex combination of two vectors \( u \) and \( v \) is defined as \( w = \alpha u + \beta v \).

a. State the two conditions that must be met by the scalars \( \alpha \) and \( \beta \) for the expression to be a convex combination. (2 points)
b. Given \( u = [1, 10] \) and \( v = [5, 2] \), calculate the vector \( w \) when \( \alpha = 0.25 \). (3 points)
c. Geometrically, where does the vector \( w \) lie in relation to \( u \) and \( v \)? (1 point)