Vectors · Study notes

MATVectors & Their Algebra

Types of Vectors, Unit Vectors & Direction

This subtopic covers the vocabulary of vectors (zero, unit, equal, collinear, coinitial, free, position vectors) and the machinery of direction: unit vectors, direction cosines, and direction ratios. A unit vector $\hat{a}=\dfrac{\vec{a}}{|\vec{a}|}$ fixes direction with magnitude $1$ , and direction cosines are the cosines of the angles a vector makes with the axes.

ISC questions ask for unit vectors, vectors of a given magnitude in a direction, and direction cosines of a vector or a joining segment.

Magnitude of a vector

|\vec{a}| = \sqrt{a_1^2 + a_2^2 + a_3^2}

\vec{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}

Unit vector in the direction of a

\hat{a} = \dfrac{\vec{a}}{|\vec{a}|}

\vec{a}\ne\vec{0}

;

\hat{a}

has magnitude

1

in the same direction.

Vector of magnitude m in a direction

\vec{v} = m\,\dfrac{\vec{a}}{|\vec{a}|}

m

is the required magnitude; the unit vector sets the direction.

Direction cosines

l = \dfrac{a_1}{|\vec{a}|},\quad m = \dfrac{a_2}{|\vec{a}|},\quad n = \dfrac{a_3}{|\vec{a}|}

l,m,n

are cosines of the angles with the

x,y,z

axes;

a_1,a_2,a_3

are the components.

Identity for direction cosines

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

Always holds for any direction; a check on computed direction cosines.

Direction cosines $l,m,n$ are exactly the components of the unit vector $\hat{a}$ .
Direction ratios are any numbers proportional to $l,m,n$ ; the components $a_1,a_2,a_3$ themselves are direction ratios.
For the segment from $A$ to $B$ , first form $\vec{AB}=\vec{b}-\vec{a}$ , then take its unit vector or direction cosines.
A unit vector always has magnitude $1$ ; multiply it by $m$ to get any vector of magnitude $m$ in that direction.
The zero vector $\vec{0}$ has no defined direction, and its direction cosines are undefined.
Equal vectors have the same magnitude and direction regardless of position; collinear (parallel) vectors are scalar multiples of one another.
Reversing direction (from $B$ to $A$ ) negates all three direction cosines.
Always verify $l^2+m^2+n^2=1$ as a quick correctness check.

Where the marks go

Using direction ratios $a_1,a_2,a_3$ directly as direction cosines without dividing by $|\vec{a}|$ .
Forgetting that direction depends on order: $\vec{AB}$ and $\vec{BA}$ have opposite-signed direction cosines.
Giving a vector of magnitude $m$ as $m\vec{a}$ instead of $m\hat{a}=m\dfrac{\vec{a}}{|\vec{a}|}$ .
Arithmetic slips in $|\vec{a}|$ that break the check $l^2+m^2+n^2=1$ , signalling an error to fix.

How the board asks it

Numericalunit vector in the direction of a
Find the unit vector in the direction of the vector $\vec{a}=2\hat{i}-3\hat{j}+6\hat{k}$ .
Numericaldirection cosines of a vector
Find the direction cosines of the vector $\vec{a}=\hat{i}+2\hat{j}+3\hat{k}$ , and verify that $l^2+m^2+n^2=1$ .
Numericalvector AB = b - a then unit vector
If $A(1,2,-1)$ and $B(4,6,11)$ , find the unit vector along $\vec{AB}$ and the direction cosines of $\vec{AB}$ .
Numericalvector of given magnitude in a direction
Find a vector of magnitude $9$ units in the direction of the vector $\vec{a}=2\hat{i}-\hat{j}+2\hat{k}$ .
Give reasonscollinear vectors as scalar multiples
Show that the vectors $\vec{a}=2\hat{i}-3\hat{j}+4\hat{k}$ and $\vec{b}=-4\hat{i}+6\hat{j}-8\hat{k}$ are collinear, and state whether they point in the same or opposite directions.
Multiple choicedirection cosines from direction ratios
The direction cosines of the vector $\hat{i}-\hat{j}+\hat{k}$ are: (a) $1,-1,1$ (b) $\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}$ (c) $\frac{1}{3},-\frac{1}{3},\frac{1}{3}$ (d) $1,1,1$

Practise this topic

NextAddition, Position Vectors & Section Formula

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.