Vector Addition, Components and Section Formula

If two vectors are represented by sides \(\vec{AB}\) and \(\vec{BC}\) of a triangle taken in order (head of one at the tail of the next), then the sum is represented by the closing side \(\vec{AC}\): \[\boxed{\;\vec{AB}+\vec{BC}=\vec{AC}\;}\]

If two vectors with the same initial point form adjacent sides of a parallelogram, their sum is the diagonal of the parallelogram drawn from the common point: \[\vec a+\vec b=\vec c\quad\text{(diagonal)}\] The two laws are equivalent — they describe the same operation.

For a scalar \(\lambda\in\mathbb R\) and vector \(\vec a\), the scalar multiple \(\lambda\vec a\) has
• magnitude \(|\lambda|\,|\vec a|\),
• direction same as \(\vec a\) if \(\lambda>0\), opposite if \(\lambda<0\), and \(\lambda\vec a=\vec 0\) if \(\lambda=0\).

Properties: \(\lambda(\vec a+\vec b)=\lambda\vec a+\lambda\vec b\); \((\lambda+\mu)\vec a=\lambda\vec a+\mu\vec a\); \(\lambda(\mu\vec a)=(\lambda\mu)\vec a\); \(1\vec a=\vec a\).

Any vector \(\vec r\) in 3-D can be uniquely written as \[\vec r=a\,\hat\imath+b\,\hat\jmath+c\,\hat k,\] where \(a, b, c\) are the scalar components (real numbers) and \(a\hat\imath, b\hat\jmath, c\hat k\) are the vector components along the three coordinate axes.

\(|\vec r|=\sqrt{a^2+b^2+c^2}\). Two vectors are equal iff all three components match.

If \(P(x_1,y_1,z_1)\) and \(Q(x_2,y_2,z_2)\) have position vectors \(\vec a,\vec b\) respectively, then \[\boxed{\;\vec{PQ}=\vec b-\vec a=(x_2-x_1)\hat\imath+(y_2-y_1)\hat\jmath+(z_2-z_1)\hat k\;}\] "Head minus tail" — direction goes FROM \(P\) TO \(Q\). Magnitude \(|\vec{PQ}|=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}\) is the distance from \(P\) to \(Q\).

If \(R\) divides segment \(PQ\) (with position vectors \(\vec a,\vec b\)) in the ratio \(m:n\),

Internal division: \(\vec{OR}=\dfrac{m\vec b+n\vec a}{m+n}\).
External division: \(\vec{OR}=\dfrac{m\vec b-n\vec a}{m-n}\).
Midpoint (\(m=n=1\)): \(\vec{OR}=\dfrac{\vec a+\vec b}{2}\).