Pure Mthematics - Vectors

Question 1

What is the vector equation of a line form

Answer 1

r = a + λd

Question 2

What is r in: r = a + λd

Answer 2

A general point on the vector line

Question 3

What is a in: r = a + λd

Answer 3

The initial point on the line

Question 4

What is λ in: r = a + λd

Answer 4

The parameter (how many iteratiions of the direction vector)

Question 5

What is d in: r = a + λd

Answer 5

The direction vector

Question 6

In 2D what direction vector does a line with gradient p/q have

Answer 6

(q p)

Question 7

How do you prove if lines are parallel

Answer 7

If they have direction vectors which are scalar multiples of each other

Question 8

How do you prove if a point lies on a line

Answer 8

• Equate the point to the general point of the line, then check for consistent values of the parameters

Question 9

How do you find a line that passes through two points

Answer 9

1- r = a + λd
2- a can be either ppoint vector
3- d is the vector that goes from a to the other

Question 10

How do you show that 2 lines represent the same line

Answer 10

1. Show that they (are parallel) go in the same direction
2. Show that the initial point on line 1 lies on a line 2

Question 11

What is a skew line and how do you prove that 2 lines are skew

Answer 11

• 2 lines in 3D that aren’t parallel and dont intersect

Question 12

What is the equation for the Scalar Product

Answer 12

a.b = ab cos θ

Question 13

How do you find the angle between two vectors one vector is going into the center point and another one out

Answer 13

180 - (cos^-1((a.b)/ab))

Question 14

How do you prove that 2 lines are perpendicular

Answer 14

If there direction vectors have scalar products of 0

Question 15

How do you find the minimum distance between a line and a point

Answer 15

Find the vector from general point in the line to the point given
Dot this vector with the direction vector of the line
Set the product equal to zero
Sub the parameter back to get vector
Use Pythagoras to find manicure of the vector

Question 16

How do you find the distance between 2 parallel points

Answer 16

Find the magnitude of the vector that goes from the starting point of one line to a general point (find the parameter by making the scalar product = 0)

Question 17

How do you find the minimum distance between 2 skew lines

Answer 17

1. Find the vector of the general point on line 1 to the general point on line 2
2. Dot this vector with both lines direction vector
3. Set dot product = 0 as it will be perpendicular
4. Solve for the parameters
5. Sub parameters back into vector equation of perpendicular line
6. find the magnitude of that line