- How do you find the angle between two vectors?
- Use the dot product formula: cos θ = (A · B) ÷ (|A||B|), then take the inverse cosine. A · B = Ax·Bx + Ay·By + Az·Bz, and |A| is the magnitude √(x² + y² + z²). If A · B = 0 the vectors are perpendicular (90°).
- How do you add two vectors geometrically?
- Draw them tip to tail: slide the second vector, keeping its length and direction, so its tail sits on the first vector's tip. The sum is the arrow from the first vector's tail to the second vector's new tip, same as adding the matching components.
- What is the magnitude of a 3D vector?
- For a vector (x, y, z), the magnitude (length) is |v| = √(x² + y² + z²) — the 3D version of Pythagoras' theorem, and the straight-line distance from the origin to the point (x, y, z).
- What does the dot product tell you?
- A · B = Ax·Bx + Ay·By + Az·Bz is a single number (a scalar). Its sign tells you the angle: positive means acute, zero means perpendicular (90°), and negative means obtuse, up to 180° when the vectors point in exactly opposite directions.
- What is the difference between vector addition and subtraction?
- A + B is built tip to tail with B as drawn. A − B is the same construction with B reversed (pointing the opposite way) — equivalently, A − B is the vector from B's tip to A's tip when both start at the origin.
- Can two vectors be both perpendicular and have different lengths?
- Yes — perpendicularity only depends on direction (the angle being 90°), not on magnitude. Try the Perpendicular preset here: A and B have different lengths but A · B = 0.