Motion in a Plane

Same magnitude AND same direction

Step 1: Two vectors are said to be equal if they satisfy two conditions simultaneously. Step 2: Condition 1 — they must have the same magnitude. Step 3: Condition 2 — they must have the same direction. Step 4: The starting point (initial point) does NOT matter — vectors can be translated freely. Step 5: If only magnitudes are equal but directions differ, the vectors are not equal (they could be negative of each other if directions are exactly opposite).

A + B

Step 1: The formula for resultant of two vectors is R = √(A² + B² + 2AB cosθ). Step 2: When θ = 0°, cos 0° = 1. Step 3: R = √(A² + B² + 2AB × 1) = √(A + B)² = A + B. Step 4: This makes physical sense — when both vectors point the same way, their effects add up fully. Step 5: The minimum resultant (A – B) occurs when θ = 180° (opposite directions). The formula √(A² + B²) is for θ = 90°.

AB cos θ

Step 1: There are two types of products of two vectors — scalar (dot) product and vector (cross) product. Step 2: The scalar product is defined as A⃗ · B⃗ = AB cos θ, where θ is the angle between them. Step 3: The result is a scalar quantity (no direction). Step 4: Example: Work = F⃗ · s⃗ = Fs cos θ. Step 5: Common confusion — AB sin θ is the magnitude of the cross (vector) product, not the dot product. Remember: Dot → cos, Cross → sin.

Zero

Step 1: Use the dot product formula: A⃗ · B⃗ = AB cos θ. Step 2: For perpendicular vectors, θ = 90°. Step 3: cos 90° = 0. Step 4: Therefore A⃗ · B⃗ = AB × 0 = 0. Step 5: Physical meaning — if force is perpendicular to displacement, work done = 0. This is why the normal force does no work on a horizontally moving object. Common mistake: students write 1 thinking of parallel vectors — that is when θ = 0°, cos 0° = 1.