Addition, Position Vectors & Section Formula

MATVectors & Their Algebra

A position vector locates a point relative to a fixed origin $O$ , written $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$ . Vector addition obeys the triangle and parallelogram laws and lets us combine displacements or forces, while the section formula gives the position vector of a point dividing a segment in a given ratio.

These tools are examined through collinearity proofs, midpoints, internal/external division, and resultant-displacement word problems.

Position vector of a point

\vec{OP} = x\hat{i} + y\hat{j} + z\hat{k} \quad \text{for } P(x,y,z)

O

is the origin;

x,y,z

are the coordinates of

P

Section formula (internal division)

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

R

divides

AB

internally in ratio

m:n

;

\vec{a},\vec{b}

are position vectors of

A,B

Section formula (external division)

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

R

divides

AB

externally in ratio

m:n

m \ne n

Midpoint formula

\vec{r} = \dfrac{\vec{a} + \vec{b}}{2}

Midpoint of

AB

; special case of internal division with

m:n = 1:1

Vector joining two points

\vec{AB} = \vec{b} - \vec{a}

\vec{a},\vec{b}

are position vectors of

A,B

; directed from

A

B

Collinearity condition

\vec{AB} = \lambda\,\vec{AC} \quad \text{for some scalar } \lambda

A,B,C

are collinear iff one joining vector is a scalar multiple of another.

To find $\vec{AB}$ , subtract position vectors in order $\vec{b} - \vec{a}$ (head minus tail); the magnitude $|\vec{AB}|$ is the distance $AB$ .
Resultant displacement is the vector sum $\vec{d_1} + \vec{d_2} + \cdots$ ; the straight-line distance is the magnitude of that resultant.
Resultant of several forces acting at a point is $\vec{F_1} + \vec{F_2} + \cdots$ , added component-wise; its magnitude is $\sqrt{F_x^2 + F_y^2 + F_z^2}$ .
For internal division in ratio $m:n$ , $R$ lies between $A$ and $B$ ; for external division $R$ lies on the extension of $AB$ .
Three points are collinear if $\vec{AB}$ and $\vec{BC}$ (or $\vec{AC}$ ) are parallel, i.e. have proportional components or $\vec{AB} = \lambda\,\vec{BC}$ .
The midpoint of $A(x_1,y_1,z_1)$ and $B(x_2,y_2,z_2)$ has position vector with components $\dfrac{x_1+x_2}{2},\,\dfrac{y_1+y_2}{2},\,\dfrac{z_1+z_2}{2}$ .
Equating components of a vector equation gives a system of scalar equations; use this to solve for an unknown such as $k$ or $\lambda$ .
Vector addition is commutative and associative: $\vec{a}+\vec{b}=\vec{b}+\vec{a}$ and $(\vec{a}+\vec{b})+\vec{c}=\vec{a}+(\vec{b}+\vec{c})$ .

Where the marks go

Swapping the weights in the section formula: for internal ratio $m:n$ , point $B$ (the far end) carries weight $m$ , giving $\dfrac{m\vec{b}+n\vec{a}}{m+n}$ , not $\dfrac{m\vec{a}+n\vec{b}}{m+n}$ .
Using $\vec{a}-\vec{b}$ instead of $\vec{b}-\vec{a}$ for $\vec{AB}$ , which reverses the direction.
Confusing internal ( $+$ in numerator) with external ( $-$ in numerator) division formulas.
Claiming collinearity from equal lengths or from one shared point; you must show the joining vectors are scalar multiples (parallel).

How the board asks it

Numericalthe section formula and midpoint formula
Find the position vector of the point $R$ which divides the line joining $A(2\hat{i}-3\hat{j}+4\hat{k})$ and $B(\hat{i}+2\hat{j}-\hat{k})$ internally in the ratio $2:3$ , and also find the position vector of the midpoint of $AB$ .
Numericalvector joining two points and external division
The position vectors of $A$ and $B$ are $3\hat{i}-2\hat{j}+\hat{k}$ and $\hat{i}+\hat{j}-2\hat{k}$ . Find $\vec{AB}$ and the position vector of the point $P$ that divides $AB$ externally in the ratio $3:1$ .
Derive / provethe collinearity condition and ratio of division
Show that the points $A(-2\hat{i}+3\hat{j}+5\hat{k})$ , $B(\hat{i}+2\hat{j}+3\hat{k})$ and $C(7\hat{i}-\hat{k})$ are collinear, and find the ratio in which $B$ divides $AC$ .
Applicationresultant of vectors added component-wise
Three forces $\vec{F_1}=2\hat{i}+\hat{j}-\hat{k}$ , $\vec{F_2}=-\hat{i}+3\hat{j}+2\hat{k}$ and $\vec{F_3}=\hat{i}-2\hat{j}+\hat{k}$ act at a point. Find the magnitude of their resultant.

Practise this topic

PreviousTypes of Vectors, Unit Vectors & Direction NextScalar (Dot) Product, Angle & Projection

Written for Sublevo. Question text quoted anywhere in these notes is the Council’s and carries its year and paper; the board’s own diagrams are not reproduced.