MH Board · Chapter

Vectors

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has

|\vec a| = \sqrt{x^2 + y^2 + z^2}

;

\overrightarrow{AB}

= position vector of B minus that of A; a unit vector is

\vec a/|\vec a|

Start

Adding vectors and the triangle lawAdd and subtract component by component; around a closed figure the vectors add to

\vec 0

; scalar multiples are parallelLocked

Collinear and coplanar vectorsThree points are collinear when

\overrightarrow{AB} = k\,\overrightarrow{AC}

; a vector is a linear combination

x\vec a + y\vec b + z\vec c

of three non-coplanar onesLocked

Section formulaInternal division in

m : n

:

\dfrac{m\vec b + n\vec a}{m + n}

; external:

\dfrac{m\vec b - n\vec a}{m - n}

; midpoint and centroid as special casesLocked

Dot product and the angle between vectors $\vec a\cdot\vec b = |\vec a||\vec b|\cos\theta = a_1b_1 + a_2b_2 + a_3b_3$ ; zero means perpendicularLocked
Projection of a vectorThe scalar projection of $\vec a$ on $\vec b$
Direction cosines and direction ratios $l = \cos\alpha$ , $m = \cos\beta$ , $n = \cos\gamma$

Cross product $\vec a\times\vec b$ is the determinant with rows $\hat i\ \hat j\ \hat k$ , $\vec a$ , $\vec b$ ; it is perpendicular to both, and $\vec b\times\vec a = -\vec a\times\vec b$ Locked
Areas and unit normals from the cross product $|\vec a\times\vec b| = |\vec a||\vec b|\sin\theta$

Scalar triple product $[\vec a\ \vec b\ \vec c] = \vec a\cdot(\vec b\times\vec c)$ , the 3 by 3 determinant of the components; cyclic swaps keep it, one swap negates itLocked
Volume and coplanarity $|[\vec a\ \vec b\ \vec c]|$
Vector triple product $\vec a\times(\vec b\times\vec c) = (\vec a\cdot\vec c)\vec b - (\vec a\cdot\vec b)\vec c$ ,
Identities with triple productsExpand brackets in $[\ ]$ , drop terms with a repeated vector: $[\vec a + \vec b\ \ \vec b + \vec c\ \ \vec c + \vec a] = 2[\vec a\ \vec b\ \vec c]$

is

\dfrac{\vec a\cdot\vec b}{|\vec b|}

;

the vector projection multiplies it by

\hat b

Locked

with

l^2 + m^2 + n^2 = 1

;

ratios

a : b : c

scale to cosines by

\dfrac{1}{\sqrt{a^2 + b^2 + c^2}}

Locked

is the parallelogram's area; half of it for a triangle;

\pm\dfrac{\vec a\times\vec b}{|\vec a\times\vec b|}

are the unit normals

Locked

is the parallelepiped's volume, one sixth of it the tetrahedron's; it is

0

exactly when the three vectors are coplanar

Locked

which lies in the plane of

\vec b

and

\vec c

Locked