Find the scalar components and magnitude of the vector joining the points P(x₁, y₁, z₁) and Q(x₂, y₂, z₂).
The vector joining points P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) is given by:
PQ = (x₂ - x₁)î + (y₂ - y₁)ĵ + (z₂ - z₁)k̂
The scalar components of this vector are (x₂ - x₁), (y₂ - y₁), and (z₂ - z₁).
The magnitude of the vector is:
|PQ| = √[(x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²]
Explanation
The context clearly explains in section 10.5.2 that when we have two points P₁(x₁, y₁, z₁) and P₂(x₂, y₂, z₂), the vector joining them is obtained by subtracting the position vectors. The scalar components are the coefficients of î, ĵ, and k̂ in the component form. The magnitude formula follows from the distance formula between two points in 3D space.
Solution Steps
Step 1: Write the position vectors of P and Q as OP = x₁î + y₁ĵ + z₁k̂ and OQ = x₂î + y₂ĵ + z₂k̂
Step 2: Find the vector PQ = OQ - OP = (x₂ - x₁)î + (y₂ - y₁)ĵ + (z₂ - z₁)k̂
Step 3: Identify scalar components as (x₂ - x₁), (y₂ - y₁), (z₂ - z₁)
Step 4: Apply magnitude formula |PQ| = √[(x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²]