A vector is defined by direction and magnitude. Vectors can be used to describe any of the physical forces (gravitational, electrical, tension, ...), as well as motion (displacement, velocity, acceleration).

In geometric representation, vectors are drawn as an arrow, and its length represents the magnitude of the force or motion it has.

The force, F, acting on an object is a vector, and can be written as $F↖{→}$, or **F** (bold type).

The magnitude of vector ${AB}↖{→}$ is written as $|AB↖{→}|$, and means the size of the vector from initial point A to terminal point B.

The magnitude of a vector is a scalar value.

Two vectors are considered equivalent if the have the same direction and magnitude: ${AB}↖{→}$ = ${XY}↖{→}$

$BA↖{→}$ is the opposite vector to ${AB}↖{→}$, since it has the same magnitude in the reverse direction.

The addition of vectors is commutative. It does not matter which order they are added, the result is the same.

**a** + **b** = **b** + **a**

**a** + **a** = 2**a**

If **a** is a unit vector, a parallel vector (pointing in the same direction) **b** can be expressed as **b** = k**a**, where $k$ is a real number.

Subtracting a vector **b** from vector **a**, is the same as adding the opposite of **b**.

Commutative: α**b** = **b**α

Associative: α(β**b**) = (αβ)**b**

Distributive: (1) α(**b** + **c**) = α**b** + α**c**

Distributive: (2) (α + β)**b** = α**b** + β**b**

Multiplicative identity: 1**b** = **b**

Zero: 0**b** = 0 and α0 = 0

A vector of length 1 unit in the direction of **a** is: ${a}/{|a|}$

Vectors in 2D space can be described by components in the x and y directions.

One unit in the positive x-axis direction is the unit vector **i**, and one unit in the positive y-axis direction is the unit vector **j**. These can be written as **i** = $(\table 1;0)$ and **j** = $(\table 0;1)$

In the graph above, ${OQ}↖{→}$ can be described as 6 units to the right, and 2 units upwards, or: ${OQ}↖{→}$ = 6**i** + 2**j**

This makes addition and subtraction of vectors much easier:

${OQ}↖{→} + {OP}↖{→}$ = (6**i** + 2**j**) + (-4**i** + 4**j**) = 2**i** + 6**j**

${OQ}↖{→} - {OP}↖{→}$ = (6**i** + 2**j**) - (-4**i** + 4**j**) = 10**i** - 2**j**

${OQ}↖{→}$ = 6**i** + 2**j** can also be written as ${OQ}↖{→}$ = $(\table 6;2)$

So ${OQ}↖{→} + {OP}↖{→}$ = $(\table 6;2)$ + $(\table -4;4)$ = $(\table 2;6)$

A 3D vector can be written as **v** = $(\table a;b;c)$, which means **v** = a**i** + b**j** + b**k**, where **i**, **j**, and **k** are the unit vectors in the x, y, and z directions respectively.

**u**⋅**v** = |**u**||**v**| cosθ

**u**⋅**v** = ±|**v**||**u**| if **u** and **v** are parallel.

**v**⋅**u** = 0 if **u** and **v** are orthogonal (90°).

**u**⋅**v** = |**u**$|^2$

**u**⋅**v** > 0 for θ < 90°, and **u**⋅**v** < 0 for θ > 90°

**u**⋅**v** = **v**⋅**u**

**u**⋅(**v** + **w**) = **u**⋅**v** + **u**⋅**w**

(λ**u**)⋅**v** = λ(**u**⋅**v**)

If **a** = $a_1$**i** + $a_2$**j** + $a_3$**k** and

**b** = $b_1$**i** + $b_2$**j** + $b_3$**k**, then:

**a**⋅**b** = $a_1b_1 + a_2b_2+ a_3b_3$

The scalar product of two vectors (or inner product or dot product) is defined as:

**u**⋅**v** = |**u**||**v**| cosθ, where **u** and **v** are two non-zero vectors, at angle θ.

**v**cosθ is the projection of **v** in the direction of **u**.

The scalar product is a number.

The scalar product is very useful for finding the angle between vectors:

$$cosθ={a⋅b}/{|a||b|}$$For perpendicular vectors **a**⋅**b** = 0

For parallel vectors **a**⋅**b** = |**a**||**b**|

For coincident vectors **a**⋅**a** = $a^2$

The vector equation of a line is **r** = **a** + $t$**b**, where **r** is the general position vector a point on the line, **a** is a given position vector of a point on the line and **b** is a direction vector parallel to the line. $t$ is the parameter.

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Environmental Science is the most important of all sciences. As the world enters a phase of climate change, unprecedented biodiversity loss, pollution and human population growth, the management of our environment is vital for our futures. Learn about Environmental Science on ScienceLibrary.info.

1695 - 1726

Nicolaus (II) Bernoulli was the second mathematician called Nicolaus in the Bernoulli dynasty, which dominated the world of mathematics from their home in Basel, Switzlerland.

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