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 one point to the other

Question 10

How do you show that 2 lines represent the same line

Answer 10

1. Show that they go in the same direction
2. Show that they have the same point

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 magnitude of the vector that goes from the point and is perpendicular to the line

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