Vectors

Question 1

Column Vectors

Answer 1

A translation can be described using column vectors.
A column vector describes the horizontal movement (x) and vertical movement (y) these are vector components.

Vectors are indicated with a wavy line underneath the letter of AB with an arrow pointing in the direction of the vector

Note
Vectors can start anywhere on the number plane

Question 2

Multiplying a vector by a scalar

Answer 2

If s is any number (scalar) and a is any vector then sa is another vector
If a = (x|y) then sa = (xs|ys)

Note s is called a scalar because it usually changes the scale of the vector

Question 3

Inverse

Answer 3

The inverse of a vector is obtained by changing the sign of the components of the vector

Note
This has the effect of changing the direction of the arrow on the vector

Question 4

Addition and subtraction

Answer 4

Vectors can be added or subtracted by performing one translation after another

If a = (x|y) and b = (p|q) then a + b = (x+p | y+q)

Note
a + b = b + a this is called the commutative rule for addition of vectors

If a = (x|y) and b = (p|q) then a – b = (x–p, y–q)

Note
Vector subtraction is best done by addition of the inverse or opposite vector

Question 5

Magnitude or modulus of a vector

Answer 5

The magnitude or size of a vector is represented by its length, the longer the length the greater the magnitude

The magnitude of a~ or AB→ is denoted by |a~| or |AB→| respectively and calculated using Pythagoras’ theorem

a = (x|y)
Magnitude of vector a |a| = √(x² + y²)

Question 6

Argument (angle) of a vector θ

Answer 6

The argument of a~ or AB→ is denoted by θ and is calculated using Pythagoras’ theorem

a = (x|y)
Argument of vector a~
tanθ = y/x
θ = tan⁻¹(y/x)

Question 7

Vectors in 3 dimensions

Answer 7

The usual convention is to take x and y axis in a horizontal plane. Now in 3 dimensions we add z axis pointing vertically upwards

Question 8

Magnitude or modulus of a vector in 3D

Answer 8

a = (x,y,z)
|a| = √(x² + y² + z²)

Question 9

Displacement and position vectors

Answer 9

Displacement can start from anywhere
Position vectors are displacement vectors that start at the origin

Question 10

Parallelogram rule of addition

Answer 10

To identify the point with the position vector OA> + OB> is not easy because the arrows from O to A and from O to B are not related in the way needed for addition

Therefore the parallelogram OACB must be completed
OA> + OB = OA> + AC> = OC>
Note
OB> = AC>

Question 11

Parallelogram rule of subtraction

Answer 11

To identify the point with position vector OB> – OC> is not easy because the arrows from O to A and from O to B are not related in the way needed for subtraction

Note
Vector subtraction is best done by addition of the inverse

Therefore the parallelogram O(–A)CB must be completed
OB> – OA> = OB> + BC> = OC> = AB>
Note
OA> = –OA> = BC> and OC> = AB>
Therefore
AB> = OB> – OA>

Question 12

Component vectors

Answer 12

Apply the rules of vector algebra to a vector in column vector form,
The vector (1|0) and (0|1) are called unit vectors in the x and y directions.
They are denoted by the letter i = (1|0) and j = (0|1)

Question 13

Component vectors in 3D

Answer 13

a = (x|y|z) and b (q|r|s) are written in component form a.b = xq + yr + zs
where i = (1|0|0) , j = (0|1|0) and
k = (0|0|1)

Question 14

Dot products (scalar products) of 2 vectors

Answer 14

If you have 2 vectors which are not identical but have equal magnitude, then you can write |a~| = |b~|
If s is a scalar multiple of a~ then it follows from the definition that sa = |s|.|a|

Note
This is true when s is positive, negative or zero

The dot products of vector a~ and b~ is a number a.b = |a|.|b|.cosθ, where θ is the angle between the directions of a and b.

Note
The angle θ may be acute or obtuse but it is important that it is the angle between a and b not the angle between a and –b

Note
For unit vectors i, j and k ii = jj = kk = 1 and ij = jk = ki = 0

Question 15

Dot products (scalar products) in component form (2D)

Answer 15

a = (x|y) and b = (p|q) are written in component form a.b = xp + yq

Question 16

Dot products (scalar products) in component form (3D)

Answer 16

a = (x|y|z) and b = (p|q|r) are written in component form a.b = xp + yq + zr

where i = (1|0|0) , j = (0|1|0) and
k = (0|0|1)

Question 17

Properties of dot products

Answer 17

s . (a~.b~) = (sa~) . b~

If s > 0 then the angle between sa~ and b~ is θ
If s < 0 then the angle between sa~ and b~ is π – θ

Commutative rule for scalar products
p~.q~ = q~.p~

Distributive rule for scalar products
(p~ + q~) . r = p~.r + q~.r

Question 18

Perpendicular (orthogonal) vectors

Answer 18

If vectors a~ and b~ are perpendicular then a.b = 0
If neither a nor b is a zero vector (0|0) and a.b = 0 then the vectors are perpendicular

Question 19

Parallel vectors

Answer 19

If vectors a~ and b~ are parallel then a~.b~ = ± |a~||b~|

Question 20

Angle between 2 vectors

Answer 20

cosθ = a~.b~ / [|a|.|b|

a.b = |a~|.|b~|.cosθ

x₁x₂ + y₁y₂ / [√(x₁)²+(y₁)² . √(x₂)²+(y₂)²

Steps:
1. Modulus of a
2. Modulus of b
3. a.b
4. solve for θ

Question 21

Unit vector of a~

Answer 21

A unit vector with magnitude 1 like i, j and k. To find the unit vector in the same direction as a given vector, divide the vector by its magnitude

Unit of vector a~
a~/|a~| = x₁i + y₁j / [√(x₁)²+(y₁)²

Question 22

Vector equation of a line in 2D

Answer 22

A vector equation of a line passing through a fixed point 𝐴 with position vector 𝑎 and parallel to a vector 𝑏 is 𝑟 = 𝑎 + 𝑡𝑏, where 𝑡 is a scalar parameter. This is called the vector equation of the line.

Note
To find the vector equation of a line, you need to have either
* A point it goes through and its direction or
* Two points it goes through.

Question 23

Pairs of lines in 2D

Answer 23

In 2D a pair of lines are either parallel or they intersect

The lines with vector equation r = a + sp and r = b + tq have the same direction if p is a multiple of q. If in addition b - a or a - b is a multiple of q then lines are the same, otherwise the lines are parallel

Note:
2 equations may represent the same line even though the vector a, b, p, q are different

Question 24

Vector equation of a line in 3D

Answer 24

The same principles involved in 2D vectors carries on into 3D vectors except for gradient (direction is used instead) and in 3D non-parallel lines may or may not meet.
If they don’t meet they are skew.

The lines with vector equation r = a + sp and r = b + tq intersect if unique values of s and t can be found such that a + sp = b + tq. If 2 unique values can not be found then the lines are skew

A vector equation of the line through 2 fixed points A and B with position vectors a and b is given by the equation r = a + t(b – a) where t is a scalar parameter.

Question 25

Distance from a point to a line

Answer 25

r = a + tp the vector p can be any vector in the direction of the line, in the case when p is a unit vector u, the equation becomes r = a + tu.
In this case |t| is the distance along the line from A to the point with parameter t.

Note: this principal is the same in 3D