Revise: Introduction to Vectors

Scalars have magnitude only; vectors have magnitude AND direction.

Mass = scalar. Force = vector.

Triangle Law: draw tip-to-tail; resultant runs first tail to last head.

3 east + 4 north → resultant 5.

MN = ½(c−b) = ½BC ⇒ MN ∥ BC, half its length.

QR=18 → MN=9.

OP = (n·OA + m·OB)/(m+n), when AP:PB = m:n.

AP:PB=2:3 → OP=(3OA+2OB)/5.

AB = OB − OA (destination minus start).

OB=−OA, |OA|=10 → |AB|=20.

|v| = √(x²+y²).

v=(4,3) → |v|=5.

→AB runs A to B; →BA is the exact reverse.

→PQ 8 east → →QP 8 west.

Subtraction: a−b = a+(−b), reverse then add.

6 east, 8 north → resultant 10.

Vector proofs replace angle-chasing with pure algebra.

AM=½b, AN=½c → MN=½(c−b).

Equal position vectors (same magnitude AND direction) ⇒ same point.

|OA|=7, OA'=2·OA → |OA'|=14.

v=(x,y): x horizontal, y vertical — always in that order.

(−3,6) → y-component 6.

x,y are right-triangle legs; |v| is the hypotenuse.

w=(6,8) → |w|=10.