Mathematics · Ch 8 — Vectors

Summary

A vector has both magnitude and direction; a scalar has magnitude only.
Direction cosines $l,m,n$ are the cosines of the angles a vector makes with the $x,y,z$ axes, and always satisfy $l^2+m^2+n^2=1$ ; direction ratios are any numbers proportional to $l,m,n$ .
Special vector types: equal (same magnitude and direction), unit ( $|\hat a|=1$ , $\hat a=\vec a/|\vec a|$ ), zero ( $\vec 0$ , undefined direction), parallel/collinear ( $\vec b=\lambda\vec a$ ).
The position vector of $P(x,y,z)$ is $\vec{OP}=x\hat i+y\hat j+z\hat k$ ; the vector joining $A$ to $B$ is $\vec{AB}=\vec{OB}-\vec{OA}$ .
Vectors add by the triangle law ( $\vec{AB}+\vec{BC}=\vec{AC}$ ) or the parallelogram law, componentwise in coordinates; scalar multiplication scales magnitude and may reverse direction.
The section formula: the point dividing $A,B$ internally in ratio $m:n$ has position vector $\dfrac{m\vec b+n\vec a}{m+n}$ (externally, $\dfrac{m\vec b-n\vec a}{m-n}$ ).
The scalar (dot) product $\vec a\cdot\vec b=|\vec a||\vec b|\cos\theta=a_1b_1+a_2b_2+a_3b_3$ is a scalar; it is zero exactly when the two vectors are perpendicular. …