Math Problem Statement

Найти длины векторов |𝑎 ⃗| и |𝑏 ⃗ ⃗|, скалярное произведение (𝑎 ⃗ , 𝑏 ⃗ ⃗), координаты нормированного вектора 𝑠 ⃗ сонаправленный вектору 𝑎 ⃗ и координаты вектора 𝑐 ⃗, у которого длина вектора равна 4 и противоположен направлен вектору 𝑏 ⃗ ⃗ , если: 𝑎 ⃗ = (3,2, −1), 𝑏 ⃗ ⃗ = (3,4,0).

Solution

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Math Problem Analysis

Mathematical Concepts

Vector Length
Dot Product
Normalized Vector
Vector Opposite Direction

Formulas

|a⃗| = √(a₁² + a₂² + a₃²)
|b⃗⃗| = √(b₁² + b₂² + b₃²)
a⃗ · b⃗⃗ = a₁b₁ + a₂b₂ + a₃b₃
s⃗ = a⃗ / |a⃗|
c⃗ = -b⃗⃗ * (4 / |b⃗⃗|)

Theorems

Properties of Dot Product
Normalization of Vectors
Opposite Direction Vector Scaling

Suitable Grade Level

Grades 10-12

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