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Vector operations

Vector Geometry

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.

Vector Algebra

Vector Addition

Vector addition
Vector addition: the arrows are placed tip to tail in addition

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 = 2a

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

Vector Subtraction

Vector subtraction
Vector subtraction: vectors are arranged tip to tip for subtraction

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

Properties of Scalar Multiplication

Commutative: αb = bα

Associative: α(βb) = (αβ)b

Distributive: (1) α(b + c) = αb + αc

Distributive: (2) (α + β)b = αb + βb

Multiplicative identity: 1b = b

Zero: 0b = 0 and α0 = 0

Unit Vectors

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: ${OQ}↖{→}$ = 6i + 2j

This makes addition and subtraction of vectors much easier:

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

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

${OQ}↖{→}$ = 6i + 2j 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 = ai + bj + bk, where i, j, and k are the unit vectors in the x, y, and z directions respectively.

Scalar (Dot) Product

Scalar Product or Dot Product

uv = |u||v| cosθ

uv = ±|v||u| if u and v are parallel.

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

uv = |u$|^2$

uv > 0 for θ < 90°, and uv < 0 for θ > 90°

uv = vu

u⋅(v + w) = uv + uw

u)⋅v = λ(uv)

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

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

ab = $a_1b_1 + a_2b_2+ a_3b_3$

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

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

vcosθ 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 ab = 0

For parallel vectors ab = |a||b|

For coincident vectors aa = $a^2$

Vector equation of a line

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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