Core Pure - Vectors

Question 1

How many different ways are there to derive a vector equation?

Answer 1

2

Question 2

What does the first way to derive a vector equation need to be done?

Answer 2

Given 1 position vector
Given 1 parallel vector to the line

Question 3

A vector equation of a straight line passing thought the point A, with position vector a, and parallel to the vector b is:

Answer 3

Question 4

What is the purpose of the scalar parameter?

Answer 4

By taking different values of the parameter, you can find the position vectors of different points on that straight line

Question 5

What do you need to derive a vector equation using the second method?

Answer 5

2 position vectors?

Question 6

A vector equation of a straight line passing through the points C and D with position vectors c and d respectively is:

Answer 6

Question 7

How can you work out the cartesian equation of a vector equation?

Answer 7

Question 8

What does “2 points are collinear” mean?

Answer 8

The 2 points are on the same straight line

Question 9

What is the vector equation of a plane

Answer 9

Question 10

What is the explanation of the vector equation of the plane?

Answer 10

Question 11

What do you need to be given to derive the vector equation of a plane?

Answer 11

A position vector in the plane
2 non parallel, non zero vectors in the plane

Question 12

What is the normal vector?
How is it usually written?

Answer 12

A vector that is perpendicular to the plane

The normal vector is usually written as n

Question 13

What is the cartesian equation of a plane in the 3D?

Answer 13

Question 14

How is a plane generally denoted?

Answer 14

A capital pi

Question 15

What does coplanar mean?

Answer 15

Points are coplanar if they all lie on the same plane

Question 16

What is the angle between 2 vectors?

Answer 16

The angle is the angle between the way that they are both pointing

Question 17

What is the scalar product often called?

Answer 17

The dot product

Question 18

How is the scalar product defined?

Answer 18

Question 19

What do you know about the dot product when 2 vectors are perpendicular?

Explain the logic

Answer 19

If 2 vectors are perpendicular then you will have cos90
Since cos90=0 the dot product of perpendicular vectors is always 0
Therefore the non zero vectors of a and b are only perpendicular if a.b=0

Question 20

What do you know about the dot product when 2 vectors are parallel?

Explain the logic

Answer 20

If 2 vectors are perpendicular then you will have cos90
Since cos90=0 the dot product of perpendicular vectors is always 0
Therefore the non zero vectors of a and b are only perpendicular if a.b=0

Question 21

What is the simple form of the dot product?

Answer 21

Question 22

Answer 22

Question 23

With all formulas in 9.4 how do you find the obtuse angle?

Answer 23

Use the formula to find the acute angle ,x, then do 180-x=y

Question 24

What is the scalar product form of a plane?

Answer 24

Question 25

Explain how we know the scalar product form of a plane?

Answer 25

Question 26

How can you convert between scalar and cartesian form for the equation of a plane?

Answer 26

Question 27

What is a common mistake that I often make? (Vectors)

Answer 27

I forget that they a lot of vectors are position vectors

Question 28

Answer 28

Question 29

Answer 29

Question 30

What is the algorithm that can be used to determine points of intersection?

Answer 30

Question 31

What does it mean if 2 lines are skew?

Answer 31

Two straight lines are said to be skew if they are not parallel **and** they do not intersect

Question 32

What 3 perpendicular distances do you need to be able to calculate?

Answer 32

Between 2 non intersecting lines Between a line and a point Between a point and a plane

Question 33

What is the method to find the perpendicular distance between two non intersecting lines?

Answer 33

For this type of question you find the general vectors OA and OB by making r equal to one vector by adding the position and direction vectors of each equation then find AB in terms of the different parameters. Now you know that the angle between line 1 and AB is 90 so you can calculate the dot product of AB.(the direction vector of line 1) which will equal 0. You can then find the dot product of AB.(the direction vector of line 2). Now you can solve the simultaneous equations for the parameters and then substitute your values back into the position vector AB. Then find the modulus of AB as the distance between then 2 lines

Question 34

What is the method to find the perpendicular distance between a line and a point?

Answer 34

This uses the same method as a line and line but there will only be 1 vector equation and therefore only 1 parameter to solve for. Since there is only 1 line you only need to do the dot product with AB.(the direction vector of that 1 line) this will allow you to solve for the parameter and therefore sub into AB and find the modulus

Question 35

What is the method to find the perpendicular distance between a point and a plane?

Answer 35

Use the formular: