Addition of Vectors and Scalar Multiplication

Triangle Law of Addition

If two vectors $\vec a=\vec{AB}$ and $\vec b=\vec{BC}$ are placed so the terminal point of the first coincides with the initial point of the second, their sum $\vec a+\vec b$ is the vector $\vec{AC}$ from the initial point of the first to the terminal point of the second:

This is the triangle law of vector addition. In particular $\vec{AB}+\vec{BA}=\vec{AA}=\vec 0$, confirming $\vec{BA}=-\vec{AB}$.

Parallelogram Law of Addition

Equivalently, if $\vec a$ and $\vec b$ are drawn from the same initial point $O$ as two adjacent sides $OA,OB$ of a parallelogram $OACB$, their sum $\vec a+\vec b$ is the diagonal $\vec{OC}$ through $O$. The triangle and parallelogram laws describe the same addition, viewed from two constructions, and both generalise the everyday idea of combining two displacements or two forces into a single resultant.

Addition in Component Form

If $\vec a=a_1\hat i+a_2\hat j+a_3\hat k$ and $\vec b=b_1\hat i+b_2\hat j+b_3\hat k$, addition and subtraction are done componentwise:

Properties of Vector Addition

For any vectors $\vec a,\vec b,\vec c$:

• Commutativity: $\vec a+\vec b=\vec b+\vec a$.
• Associativity: $(\vec a+\vec b)+\vec c=\vec a+(\vec b+\vec c)$.
• Additive identity: $\vec a+\vec 0=\vec a$.
• Additive inverse: $\vec a+(-\vec a)=\vec 0$.

Multiplication of a Vector by a Scalar

For a vector $\vec a$ and a real number $\lambda$, the product $\lambda\vec a$ is the vector of magnitude $|\lambda|\,|\vec a|$, with the same direction as $\vec a$ when $\lambda>0$, opposite direction when $\lambda<0$, and equal to $\vec 0$ when $\lambda=0$. In component form,

Combined Expressions …