vector addition and scalar multiplication
If you walk three blocks east and then four blocks north, where do you end up relative to the start? Adding the two displacement vectors answers exactly that. Vector addition combines two journeys into the single straight-line journey with the same net effect. Scalar multiplication does something different: it stretches, shrinks, or reverses a single vector — going 'twice as far the same way', or 'the same distance but backwards'.
Geometrically, to add u and v, slide v so its tail sits on the head of u; the arrow from u's tail to v's head is the sum u + v. That is the triangle law (placing them head-to-tail). Equivalently the parallelogram law: draw u and v from a common point, complete the parallelogram, and the diagonal is u + v. In components addition is wonderfully simple — just add matching entries: (a_1, b_1, c_1) + (a_2, b_2, c_2) = (a_1 + a_2, b_1 + b_2, c_1 + c_2). Scalar multiplication by a number t scales every component: t(a, b, c) = (ta, tb, tc). If t > 1 the arrow lengthens, if 0 < t < 1 it shortens, if t < 0 it reverses direction, and its length scales by |t|, since |t v| = |t| |v|. Subtraction is addition of the reverse: u - v = u + (-1)v.
These two operations are the whole foundation: every vector v = a i + b j + c k is built by scaling the basis vectors and adding. They obey the friendly rules you expect — addition is commutative (u + v = v + u) and associative, and scaling distributes over addition. A common pitfall: |u + v| is NOT in general |u| + |v|. Two arrows pulling in different directions partly cancel, so the resultant is usually shorter than the sum of the lengths; equality holds only when u and v point the exact same way. This is the triangle inequality wearing vector clothes.
Let u = (3, 0, 0) and v = (0, 4, 0). Then u + v = (3, 4, 0), an arrow whose length is sqrt(3^2 + 4^2) = 5 — shorter than |u| + |v| = 3 + 4 = 7, because the two pull at right angles. Scaling: 2u = (6, 0, 0) is twice as long the same way; -v = (0, -4, 0) is v reversed.
Head-to-tail addition; the resultant can be shorter than the parts.
You can add a vector to a vector and a point plus a vector (which lands on a new point), but you cannot add a point to a point — that has no geometric meaning. Keep the two species apart.