Addition and Scalar Multiplication

Triangle law: \(\overrightarrow{AB}+\overrightarrow{BC}=\overrightarrow{AC}\). Parallelogram law: the diagonal from the common initial point gives the sum. Addition is commutative and associative, and \(\vec a+(-\vec a)=\vec0\).

Scalar multiplication: \(\lambda\vec a\) has magnitude \(|\lambda||\vec a|\) and the same direction as \(\vec a\) if \(\lambda>0\) (opposite if \(\lambda<0\)).

In components, add or scale coordinate-wise: \((a_1\hat i+a_2\hat j+a_3\hat k)+(b_1\hat i+b_2\hat j+b_3\hat k)=(a_1+b_1)\hat i+(a_2+b_2)\hat j+(a_3+b_3)\hat k\).

Two vectors are collinear iff \(\vec b=\lambda\vec a\), i.e. their components are proportional.