Spatial Coordinates & Vectors

Master 3D coordinate systems, vector operations, dot product, cross product, and applications in physics and engineering.

11th Grade

Vectors

~60 min

🎮 Interactive Activity: 3D Coordinate Locator

Identify the 3D coordinates!

Point P has coordinates:

x = 3, y = 4, z = 5

Write as ordered triple (x, y, z):

🎮 Interactive Activity: Vector Calculator

Perform vector operations!

Operation: Addition

v₁ = (3, 4, 0)

v₂ = (1, 2, 3)

v₁ + v₂ = ?

1. 3D Coordinate System

Understanding 3D Coordinates

In 3D space, we use three perpendicular axes: x-axis (horizontal), y-axis (vertical), and z-axis (depth). A point is represented as (x, y, z).

Example 1: Locate Point

Problem: Locate point P(3, 4, 5) in 3D space

Step 1: Move 3 units along x-axis

Step 2: Move 4 units along y-axis

Step 3: Move 5 units along z-axis

Answer: Point P is at (3, 4, 5)

Example 2: Distance from Origin

Problem: Find distance from origin to P(2, 3, 6)

Step 1: Use distance formula: d = √(x² + y² + z²)

Step 2: d = √(2² + 3² + 6²) = √(4 + 9 + 36)

Step 3: d = √49 = 7

Answer: Distance = 7 units

2. Points and Distance in 3D

3. Vector Definition and Representation

4. Vector Operations (Addition, Subtraction, Scalar Multiplication)

5. Dot Product (Scalar Product)

6. Cross Product (Vector Product)

7. Vector Magnitude and Direction

8. Real-World Applications

Frequently Asked Questions

Practice Time!

Practice Quiz

Questions

Correct

Score

What are the coordinates of point P(3, 4, 5) in 3D space?

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If v = (2, 3, 1) and w = (1, 2, 3), what is v + w?

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What is the dot product of u = (1, 2, 3) and v = (4, 5, 6)?

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What is the magnitude (length) of vector v = (3, 4, 0)?

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If v = (5, 2, 1) and w = (2, 1, 3), what is v - w?

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What is the distance between points P(1, 2, 3) and Q(4, 6, 7)?

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If u = (2, 3, 4) and v = (1, 1, 1), what is 2u - v?

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What is the dot product of perpendicular vectors?

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If v = (a, b, c), what is the magnitude |v|?

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What is the unit vector in the direction of v = (3, 4, 0)?

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