Vectors AI HL

I. The Concept of a Vector

(a) Scalar and vector quantities

Scalar: magnitude only (e.g., speed, mass, temperature).
Vector: magnitude and direction (e.g., velocity, force, displacement).

Example: 60 km/h is a scalar (speed); 60 km/h east is a vector (velocity).

(b) Directed line segments, vectors, and displacement vectors

A directed line segment from A to B represents the displacement from A to B.
The corresponding vector is written as $overrightarrow{AB}$ .

Example: If A = (2, 1) and B = (7, 4), then
$overrightarrow{AB} = begin{bmatrix}7-2 \ 4-1end{bmatrix} = begin{bmatrix}5 \ 3end{bmatrix}$ .

(c) The magnitude of a vector

For $vec{v}=begin{bmatrix}x \ yend{bmatrix}$ , its magnitude (length) is
$lvert vec{v}rvert=sqrt{x^{2}+y^{2}}$ .

Example: For $begin{bmatrix}5 \ 3end{bmatrix}$ ,
$lvert vec{v}rvert=sqrt{5^{2}+3^{2}}=sqrt{34}approx 5.83$ .

(d) Equal and opposite vectors

Equal vectors have the same magnitude and direction.
Opposite vectors have the same magnitude but opposite direction.

Example: $begin{bmatrix}4 \ 2end{bmatrix}$ and
$begin{bmatrix}4 \ 2end{bmatrix}$ are equal;
$-begin{bmatrix}4 \ 2end{bmatrix}=begin{bmatrix}-4 \ -2end{bmatrix}$ is opposite.

II. Vector Algebra and Geometrical Applications

(a) Journeys and the triangle rule

Successive displacements add head-to-tail (triangle rule):
$;vec{r}=vec{a}+vec{b};$ .

Example: Walk 3 km east then 4 km north:
$vec{a}=begin{bmatrix}3 \ 0end{bmatrix},;vec{b}=begin{bmatrix}0 \ 4end{bmatrix},;vec{r}=begin{bmatrix}3 \ 4end{bmatrix}$ , and
$lvert vec{r}rvert=sqrt{3^{2}+4^{2}}=5text{ km}$ .

(b) Using parallelograms to add and subtract vectors

Addition: Place tails together, complete the parallelogram; the diagonal is
$vec{a}+vec{b}$ .
Subtraction: $vec{a}-vec{b}=vec{a}+(-vec{b})$ .

Example: $begin{bmatrix}2 \ 1end{bmatrix}+begin{bmatrix}1 \ 3end{bmatrix}=begin{bmatrix}3 \ 4end{bmatrix}$ ;
$begin{bmatrix}2 \ 1end{bmatrix}-begin{bmatrix}1 \ 3end{bmatrix}=begin{bmatrix}1 \ -2end{bmatrix}$ .

(c) Multiplying vectors by scalars: parallel vectors

For scalar $k$ , $kvec{v}$ stretches/shrinks $vec{v}$ .
If $k>0$ the direction is unchanged; if $k<0$ the direction reverses.
Vectors are parallel iff one is a scalar multiple of the other.

Example: $3begin{bmatrix}2 \ 1end{bmatrix}=begin{bmatrix}6 \ 3end{bmatrix}$ (same direction);
$-2begin{bmatrix}2 \ 1end{bmatrix}=begin{bmatrix}-4 \ -2end{bmatrix}$ (opposite direction).

III. The Component Form of a Vector

(a) Linear combination of vectors: $;vec{c}=lambdavec{a}+muvec{b};$

Any vector in a plane can be expressed as a linear combination of two non-parallel vectors.

Example: If $vec{a}=begin{bmatrix}1 \ 0end{bmatrix}$ and
$vec{b}=begin{bmatrix}0 \ 1end{bmatrix}$ , then
$vec{c}=3vec{a}+2vec{b}=begin{bmatrix}3 \ 2end{bmatrix}$ .

(b) The unit vectors $;vec{imath};$ and $;vec{jmath};$

$vec{imath}=langle 1,0rangle$ (x-axis),
$vec{jmath}=langle 0,1rangle$ (y-axis). Any vector
$vec{v}=langle x,yrangle=xvec{imath}+yvec{jmath}$ .

Example: $langle 3,2rangle=3vec{imath}+2vec{jmath}$ .

(c) Algebra of vectors in component form

Add/subtract and scale componentwise:
$begin{bmatrix}x_{1} \ y_{1}end{bmatrix}+begin{bmatrix}x_{2} \ y_{2}end{bmatrix}=begin{bmatrix}x_{1}+x_{2} \ y_{1}+y_{2}end{bmatrix}$ ,
$kbegin{bmatrix}x \ yend{bmatrix}=begin{bmatrix}kx \ kyend{bmatrix}$ .

Example: $begin{bmatrix}2 \ 1end{bmatrix}+begin{bmatrix}-1 \ 4end{bmatrix}=begin{bmatrix}1 \ 5end{bmatrix}$ ;
$2begin{bmatrix}3 \ -2end{bmatrix}=begin{bmatrix}6 \ -4end{bmatrix}$ .

(d) The magnitude of vectors in component form

For $vec{v}=begin{bmatrix}x \ yend{bmatrix}$ ,
$lvert vec{v}rvert=sqrt{x^{2}+y^{2}}$ .

Example: $begin{bmatrix}3 \ 4end{bmatrix}$ has
$lvert vec{v}rvert=sqrt{3^{2}+4^{2}}=5$ .

IV. Unit Vectors

(a) Scaling to make a unit vector

A unit vector has magnitude 1. To find the unit vector in the direction of
$vec{v}neqvec{0}$ , divide by its magnitude:
$;hat{vec{v}}=dfrac{vec{v}}{lvert vec{v}rvert};$ .

Example: For $vec{v}=begin{bmatrix}3 \ 4end{bmatrix}$ ,
$lvert vec{v}rvert=5$ and
$hat{vec{v}}=begin{bmatrix}tfrac{3}{5} \ tfrac{4}{5}end{bmatrix}$ .

Practice Problems (click to expand)

Find the displacement vector from A(–2, 1) to B(4, 5) in column form, and calculate its magnitude.
A journey consists of walking $begin{bmatrix}6 \ 0end{bmatrix}$ then $begin{bmatrix}0 \ -8end{bmatrix}$ . Find the resultant vector and its magnitude.
If $vec{a}=begin{bmatrix}2 \ 3end{bmatrix}$ , calculate:
(i) $3vec{a}$ ,
(ii) $-tfrac{1}{2}vec{a}$ .
Express $vec{v}=begin{bmatrix}7 \ -2end{bmatrix}$ in terms of $vec{imath}$ and $vec{jmath}$ .
Find the unit vector in the direction of $vec{v}=begin{bmatrix}-5 \ 12end{bmatrix}$ .