Vectors

Question 1

Column Vectors

Answer 1

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, x on top and y below

Note
Vectors can start anywhere on the number plane

Question 2

Multiplying a vector by a scalar

Answer 2

If a = (x|y) then sa = (xs|ys)

Question 3

Inverse for vectors

Answer 3

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)

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

Question 4

Magnitude or modulus of a vector

Answer 4

The magnitude of a~ or AB→ is denoted by |a~| or |AB→|

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

Question 5

Argument (angle) of a vector θ

Answer 5

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 6

Magnitude or modulus of a vector in 3D

Answer 6

a = (x,y,z) or (i, j, k)
|a| = √(x² + y² + z²)

Question 7

Displacement and position vectors

Answer 7

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

Question 8

Parallelogram rule of addition

Answer 8

OA> + OB = OA> + AC> = OC>
Note
OB> = AC>

Question 9

Parallelogram rule of subtraction

Answer 9

AB> = OB> – OA>

Question 10

Component vectors in 3D

Answer 10

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 11

Dot products (scalar products) of 2 vectors

Answer 11

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|

Formula given

Question 12

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

Answer 12

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

Question 13

Perpendicular (orthogonal) vectors

Answer 13

If vectors a~ and b~ are perpendicular then a.b = 0

Question 14

Parallel vectors

Answer 14

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

Question 15

Angle between 2 vectors

Answer 15

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 16

Unit vector of a~

Answer 16

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 17

Vector equation of a line in 2D

Answer 17

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 18

Pairs of lines in 2D

Answer 18

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

example on pg 24

Question 19

Vector equation of a line in 3D

Answer 19

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 20

Distance from a point to a line

Answer 20

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