Pure 1.9 - Vectors

Question 1

How is the equation of a straight line written in vector form?

Answer 1

A vector equation of a straight line passing through the point 𝐴 with position vector 𝐚, and parallel to the vector 𝐛 is
𝐫=𝐚+λ𝐛
where λ is a scalar parameter.
𝐫 is the position vector of a general point on the line.

Question 2

How is the vector equation of a straight line passing through points 𝐴 and 𝐵 written?

Answer 2

A vector equation of a straight line passing through the points 𝐴 and 𝐵, with position vectors 𝐚 and 𝐛 respectively, is
𝐫=𝐚+λ(𝐛-𝐚)
where λ is a scalar parameter.
Though any point on the line can be used as the initial point in the vector equation, so 𝐫=𝐛+λ(𝐚-𝐛) is also valid.

Question 3

How can the vector equation of a straight line be converted to cartesian form?

Answer 3

If 𝐚=𝑎₁𝐢+𝑎₂𝐣+𝑎₃𝐤 and 𝐛=𝑏₁𝐢+𝑏₂𝐣+𝑏₃𝐤, the equation of the line with vector equation 𝐫=𝐚+λ𝐛 can be given in Cartesian form as
(𝑥-𝑎₁)/𝑏₁=(𝑦-𝑎₂)/𝑏₂=(𝑧-𝑎₃)/𝑏₃
Each of the three equations is equal to λ.

Question 4

How can the vector equation of a plane in three dimensions be written?

Answer 4

The vector equation of a plane can be written as
𝐫=𝐚+λ𝐛+μ𝐜
Where 𝐫 is the position vector of a general point in the plane, 𝐚 is the position vector of a point in the plane, 𝐛 and 𝐜 are both non-parallel, non-zero vectors in the plane and λ and μ are scalars.

Question 5

How can the equation of a plane in three dimensions be written in Cartesian form?

Answer 5

A Cartesian equation of a plane in three dimensions can be written in the form 𝑎𝑥+𝑏𝑦+𝑐𝑧=𝑑, where 𝑎, 𝑏, 𝑐 and 𝑑 are constants and 𝐧=𝑎𝐢+𝑏𝐣+𝑐𝐤 is the normal vector to the plane, the result can be derived using the scalar product.
Planes are often represented using the capital greek letter pi, Π.

Question 6

What is the scalar product of two vectors?

Answer 6

The scalar product of two vectors, 𝐚 and 𝐛 is written as 𝐚.𝐛 and is defined as 𝐚.𝐛=|𝐚||𝐛|cos(θ) where θ is the angle between 𝐚 and 𝐛.
If two vectors 𝐚 and 𝐛 are the position vectors of the points 𝐴 and 𝐵, then cos(∠𝐴𝑂𝐵)=𝐚.𝐛/|𝐚||𝐛|
The non-zero vectors 𝐚 and 𝐛 are perpendicular only if 𝐚.𝐛=0, since cos(90)=0
If 𝐚 and 𝐛 are parallel, 𝐚.𝐛=|𝐚||𝐛|. in particular 𝐚.𝐚=|𝐚|²
The scalar product is often called the dot product, you would say ‘𝐚 dot 𝐛’
Another method to find 𝐚.𝐛 is if 𝐚=𝑎₁𝐢+𝑎₂𝐣+𝑎₃𝐤 and 𝐛=𝑏₁𝐢+𝑏₂𝐣+𝑏₃𝐤, then 𝐚.𝐛=𝑎₁𝑏₁+𝑎₂𝑏₂+𝑎₃𝑏₃

Question 7

Describe the method of calculating the angle between a line and a plane.

Answer 7

The acute angle θ between two intersecting straight lines is given by cos(θ)=|𝐚.𝐛/|𝐚||𝐛|| where 𝐚 and 𝐛 are the direction vectors of the lines.
Suppose a plane Π passes through a given point 𝐴, with position vector 𝐚, and that the normal vector 𝐧 is perpendicular to the plane. Let 𝑅 be an arbitrary point on the plane with position vector 𝐫.
The scalar product form of the equation of a plane is 𝐫.𝐧=𝑘 where 𝑘=𝐚.𝐧 for any point in the plane with position vector 𝐚.
So the acute angle θ between the line with equation 𝐫=𝐚+λ𝐛 and the plane with equation 𝐫.𝐧=𝑘 is given by the formula
sin(θ)=|𝐛.𝐧/|𝐛||𝐧||

Question 8

Describe the process for determining whether two lines intersect, and if so where.

Answer 8

Write the equations in column notation and set them equal to each other.
Write three linear equations involving the parameters λ and μ.
Try to solve the first two equations simultaneously.
If there aren’t any solutions then the lines don’t intersect.
If the lines do intersect then check whether these solutions also satisfy the third equation.
If they don’t there is no intersection.
If they do there is an intersection, to find where this is; substitute the values of λ and μ into the equation of one of the lines.

Question 9

What is meant by two lines being skew?

Answer 9

Two lines are skew if they are not parallel and don’t intersect.

Question 10

Describe the perpendicular distance between two lines.

Answer 10

In all cases, the perpendicular distance is the shortest distance between them.
For any two non-intersecting lines 𝑙₁ and 𝑙₂, there is a unique segment 𝐴𝐵 such that 𝐴 lies on 𝑙₁, 𝐵 lies on 𝑙₂ and 𝐴𝐵 is the perpendicular between both lines.

Question 11

Describe the perpendicular distance between a point and a line.

Answer 11

The perpendicular distance from a point 𝑃 to a line 𝑙 is a line through 𝑃 which meets 𝑙 at right angles.

Question 12

Describe the perpendicular distance between a point and a plane.

Answer 12

The perpendicular distance from a point 𝑃 to a plane Π is a line through 𝑃 which is parallel to the normal vector of the plane, 𝐧.

Question 13

What is the meaning of the constant, 𝑘, in the scalar product form of the vector equation of a plane?

Answer 13

𝑘 is the length of the perpendicular from the origin to a plane Π and is written in the form 𝐫.𝐧̂=𝑘, where 𝐧̂ is the unit vector perpendicular to Π.
The perpendicular distance from the point with coordinates (α,β,γ) to the plane with the equation 𝑎𝑥+𝑏𝑦+𝑐𝑧=𝑑 is |α𝑎+β𝑏+γ𝑐-𝑑|/√(𝑎²+𝑏²+𝑐²)