The vector: a length and a direction

Space geometry and vectors, grade 12 (4 units) Lesson 7 of 20Learning

The vector: a length and a direction

A vector

A vector is a quantity with a length and a direction: walking 3 km north, the wind’s velocity, a force pulling a cart. It is drawn as an arrow – a directed segment, from its start to its end.

The vector from $\displaystyle A$ to $\displaystyle B$ is written $\displaystyle \overrightarrow{AB}$ , or with one letter: $\displaystyle \vec{u}$ .
The length of the vector $\displaystyle |\overrightarrow{AB}|$ is the length of the segment $\displaystyle AB$ .
The zero vector $\displaystyle \vec{0}$ : its start is also its end, as in $\displaystyle \overrightarrow{AA}$ . Its length is 0, and it has no direction.

Vectors in the plane and in space behave the same way: everything here holds in both.

Equal, opposite and collinear vectors

Equal vectors – the same length and the same direction, wherever they are drawn. In the parallelogram $\displaystyle ABCD$ : $\displaystyle {\overrightarrow{AB}}={\overrightarrow{DC}}$ .
Opposite vectors – the same length and opposite directions: $\displaystyle {\overrightarrow{CB}}={-\overrightarrow{AD}}$ . In particular $\displaystyle {\overrightarrow{BA}}={-\overrightarrow{AB}}$ .
Collinear vectors – on one line or on parallel lines, in the same direction or in opposite ones. $\displaystyle \overrightarrow{AD}$ and $\displaystyle \overrightarrow{CB}$ are collinear.

A vector can be moved elsewhere without changing its length or its direction – it stays the same vector.

Example: vectors in a box

In the box $\displaystyle ABCDA'B'C'D'$ :

$\displaystyle {\overrightarrow{AB}}={\overrightarrow{DC}}={\overrightarrow{A'B'}}={\overrightarrow{D'C'}}$ – four parallel equal edges, in one direction.
$\displaystyle {\overrightarrow{CD}}={-\overrightarrow{AB}}$ – the same length, the opposite direction.
All these vectors are collinear.
$\displaystyle \overrightarrow{AB}$ and $\displaystyle \overrightarrow{AD}$ have the same length only in a box whose base is a square – and even then they are not equal: their directions differ.

Exercise 1

$\displaystyle ABCDEF$ is a regular hexagon, and $\displaystyle O$ its centre. Using the letters in the drawing, write:

every vector equal to $\displaystyle \overrightarrow{AB}$

every vector opposite to $\displaystyle \overrightarrow{AB}$

a vector collinear with $\displaystyle \overrightarrow{AB}$ and twice as long

Exercise 2

In the box $\displaystyle ABCDA'B'C'D'$ : $\displaystyle {AB}={4}$ , $\displaystyle {AD}={3}$ , $\displaystyle {AA'}={12}$ .

Write every vector equal to $\displaystyle \overrightarrow{AA'}$ .

\displaystyle |\overrightarrow{AC}|

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

\displaystyle |\overrightarrow{AC'}|

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 3

True or false? Explain.

Two equal vectors have the same length.

Two vectors of the same length are equal.

Two opposite vectors are collinear.

Exercise 4

$\displaystyle ABCD$ is a quadrilateral, and $\displaystyle {\overrightarrow{AB}}={\overrightarrow{DC}}$ .
Prove that $\displaystyle ABCD$ is a parallelogram.

Formulas: space geometry and vectors, grade 12 (4 units)

Open as a page

The right-angled triangle and a triangle's area

Pythagoras’ theorem – $\displaystyle c$ the hypotenuse
$\displaystyle a^2+b^2$ $\displaystyle =$ $\displaystyle c^2$
$\displaystyle a$ the leg opposite the angle $\displaystyle \alpha$ , $\displaystyle c$ the hypotenuse
$\displaystyle \sin\alpha$ $\displaystyle =$ $\displaystyle \frac{a}{c}$
$\displaystyle b$ the leg next to the angle $\displaystyle \alpha$
$\displaystyle \cos\alpha$ $\displaystyle =$ $\displaystyle \frac{b}{c}$
The leg opposite the angle over the leg next to it
$\displaystyle \tan\alpha$ $\displaystyle =$ $\displaystyle \frac{a}{b}$
The cosine of an obtuse angle
$\displaystyle \cos(180^{\circ}-\alpha)$ $\displaystyle =$ $\displaystyle -\cos\alpha$
A triangle’s area – two sides and the angle between them
$\displaystyle S$ $\displaystyle =$ $\displaystyle \frac{1}{2}ab\sin\gamma$

The prism

The surface area – the lateral surface and the two bases; $\displaystyle B$ the base’s area
$\displaystyle S$ $\displaystyle =$ $\displaystyle M+2B$
The lateral area of a right prism – $\displaystyle P$ the base’s perimeter, $\displaystyle h$ the height
$\displaystyle M$ $\displaystyle =$ $\displaystyle P\cdot h$
The volume of a prism – $\displaystyle h$ the height, the distance between the bases
$\displaystyle V$ $\displaystyle =$ $\displaystyle B\cdot h$

The pyramid

The surface area – the lateral surface and the base
$\displaystyle S$ $\displaystyle =$ $\displaystyle M+B$
The volume of a pyramid – a third of the prism with the same base and height
$\displaystyle V$ $\displaystyle =$ $\displaystyle \frac{B\cdot h}{3}$

Vectors, geometrically

The triangle rule
$\displaystyle \overrightarrow{AB}+\overrightarrow{BC}$ $\displaystyle =$ $\displaystyle \overrightarrow{AC}$
The difference of two vectors leaving the same point
$\displaystyle \overrightarrow{AD}-\overrightarrow{AB}$ $\displaystyle =$ $\displaystyle \overrightarrow{BD}$
Collinear vectors
$\displaystyle \vec{v}=t\cdot\vec{u}$
A linear combination of $\displaystyle \vec{u}$ and $\displaystyle \vec{v}$
$\displaystyle \vec{w}=s\vec{u}+t\vec{v}$

Vectors in coordinates

The vector between two points – the end minus the start
$\displaystyle \overrightarrow{AB}$ $\displaystyle =$ $\displaystyle (x_B-x_A,\ y_B-y_A,\ z_B-z_A)$
A vector’s length
$\displaystyle |\vec{u}|$ $\displaystyle =$ $\displaystyle \sqrt{x^2+y^2+z^2}$
The midpoint of the segment $\displaystyle AB$
$\displaystyle M=\left(\frac{x_A+x_B}{2},\frac{y_A+y_B}{2},\frac{z_A+z_B}{2}\right)$

The scalar product

The definition – $\displaystyle \alpha$ the angle between the vectors
$\displaystyle \vec{u}\cdot\vec{v}$ $\displaystyle =$ $\displaystyle |\vec{u}|\cdot|\vec{v}|\cdot\cos\alpha$
In coordinates
$\displaystyle \vec{u}\cdot\vec{v}$ $\displaystyle =$ $\displaystyle x_1x_2+y_1y_2+z_1z_2$
The angle between two vectors
$\displaystyle \cos\alpha$ $\displaystyle =$ $\displaystyle \frac{\vec{u}\cdot\vec{v}}{|\vec{u}|\cdot|\vec{v}|}$
A vector’s length by the scalar product
$\displaystyle |\vec{u}|$ $\displaystyle =$ $\displaystyle \sqrt{\vec{u}^2}$
A short scalar product
$\displaystyle (\vec{u}+\vec{v})^2$ $\displaystyle =$ $\displaystyle \vec{u}^2+2\vec{u}\cdot\vec{v}+\vec{v}^2$
A short scalar product
$\displaystyle (\vec{u}+\vec{v})\cdot(\vec{u}-\vec{v})$ $\displaystyle =$ $\displaystyle \vec{u}^2-\vec{v}^2$
Perpendicular vectors
$\displaystyle \vec{u}\perp\vec{v}\iff\vec{u}\cdot\vec{v}=0$