For Two Vectors to Be Equal — Same Magnitude and Direction

Two vectors A and B are equal (A = B) if and only if:

1. |A| = |B| — same magnitude (length)
2. Direction of A = Direction of B — same direction

Both conditions must be satisfied simultaneously.

Important points: • Two vectors CAN be equal even if they are at different positions in space • Position of origin (tail) does not matter — only magnitude and direction matter • This is called the property of 'free vectors' — they can be translated freely

Not equal if: • Same magnitude, different direction → NOT equal (e.g., 5 m North ≠ 5 m South) • Same direction, different magnitude → NOT equal (e.g., 5 m North ≠ 3 m North) • Both different → clearly NOT equal

Example: • Vector A = 5 m at 30° North of East • Vector B = 5 m at 30° North of East • A = B ✓ (same magnitude 5 m AND same direction 30° N of E)

Negative vectors: • If B = −A, then B has the same magnitude as A but opposite direction • A ≠ B in this case (different directions) • A + B = 0 (they cancel each other)

Vector vs Scalar: • Scalar: has only magnitude (e.g., mass, speed, temperature) • Vector: has both magnitude AND direction (e.g., displacement, velocity, force)

Notation: • Bold: A or with arrow: →A • Magnitude: |A| or A (italic)

Components: For vectors in component form: A = (Ax, Ay) and B = (Bx, By) A = B if and only if: Ax = Bx AND Ay = By

For 3D: A = (Ax, Ay, Az) = B = (Bx, By, Bz) → Ax = Bx, Ay = By, Az = Bz (all three components must be equal)

Examples of equal vectors: • Displacement vectors: same distance and direction (any two parallel arrows of same length) • Velocity: same speed and same direction

Examples of unequal vectors: • 60 km/h East ≠ 60 km/h North (same speed, different direction) • 100 N upward ≠ 80 N upward (same direction, different magnitude) • 50 N right ≠ 50 N up (same magnitude, different direction)