Angle between Lines

Scalar Product

The scalar product is an important link between the algebra of vectors and the trigonometry of vectors. We shall see that the scalar product is somewhat comparable to the operation of multiplication on real numbers.

What is the scalar (dot) product?

The scalar product between two vectors a and b is represented by $a \cdot b$
- This is also called the dot product because of the symbol used
The scalar product between two vectors $a = a_{1} i + a_{2} j + a_{3} k$ and $b = b_{1} i + b_{2} j + b_{3} k$ is defined as $a \cdot b = a_{1} b_{1} + a_{2} b_{2} + a_{3} b_{3}$
The result of taking the scalar product of two vectors is a real number
- i.e. a scalar
For example,

$(3 i - k) \cdot (2 i + 9 j + k) = 3 \times 2 + 0 \times 9 + (- 1) \times 1 = 6 + 0 - 1 = 5$

and

$(\begin{matrix} 2 \\ 7 \end{matrix}) \cdot (\begin{matrix} - 8 \\ 2 \end{matrix}) = 2 \times (- 8) + 7 \times 2 = - 16 + 14 = - 2$

The scalar product has some important properties:
- The order of the vectors doesn’t affect the result:

$a \cdot b = b \cdot a$

In effect we can ‘multiply out’ brackets:

$a \cdot (b + c) = a \cdot b + a \cdot c$

This means that we can do many of the same things with vectors as we can do when operating on real numbers – for example,

$(a - b) \cdot (a - b) = a \cdot a - 2 a \cdot b + b \cdot b$

The scalar product between a vector and itself is equal to the square of its magnitude:

$a \cdot a = {|a|}^{2}$

For example,

$(\begin{matrix} 2 \\ 7 \end{matrix}) \cdot (\begin{matrix} 2 \\ 7 \end{matrix}) = 2^{2} + 7^{2} = 53$ and ${|(\begin{matrix} 2 \\ 7 \end{matrix})|}^{2} = 2^{2} + 7^{2} = 53$

What is the connection between the scalar product and trigonometry?

There is another important method for finding $a \cdot b$ involving the angle between the two vectors $θ$ :

$a \cdot b = |a| |b| \cos θ$

- Here $θ$ is the angle between the vectors when they are placed ‘base to base’
  - when the vectors are placed so that they begin at the same point
- This formula can be derived using the cosine rule and expanding $(a - b) \cdot (a - b)$
The scalar product of two vectors gives information about the angle between the two vectors
- If the scalar product is positive then the angle between the two vectors is acute (less than 90°)
- If the scalar product is negative then the angle between the two vectors is obtuse (between 90° and 180°)
- If the scalar product is zero then the angle between the two vectors is 90° (the two vectors are perpendicular)

7-3-3-the-scalar-product

How do I tell if vectors or lines are perpendicular?

Two (non-zero) vectors $a$ and $b$ are perpendicular if, and only if, $a \cdot b = 0$
- If the a and b are perpendicular then:
  - $θ = 90 ° \Rightarrow \cos θ = 0 \Rightarrow |a| |b| \cos θ = 0 \Rightarrow a \cdot b = 0$
- If $a \cdot b = 0$ then:
  - $|a| |b| \cos θ = 0 \Rightarrow \cos θ = 0 \Rightarrow θ = 90 ° \Rightarrow$ a and b are perpendicular
- For example, the vectors $2 i - 3 j + 5 k$ and $- 4 i - j + k$ are perpendicular since

$(2 i - 3 j + 5 k) \cdot (- 4 i - j + k) = 2 \times (- 4) + (- 3) \times (- 1) + 5 \times 1 = - 8 + 3 + 5 = 0$

Examiner Tip

When writing a scalar product, it’s important to write a distinctive dot between the vectors – otherwise your meaning will not be clear.

Worked example

Find the value of t such that the two vectors $v = (\binom{2}{\begin{matrix} t \\ 5 \end{matrix}})$ and $w = (t - 1) i - j + k$ are perpendicular to each other.

3-9-4-ib-aa-hl-the-angle-between-vectors-we-solution

How do I find the angle between two vectors?

Recall that a formula for the scalar (or ‘dot’) between vectors $a$ and $b$ is

$a \cdot b = |a| |b| \cos θ$

- where $θ$ is the angle between the vectors when they are placed ‘base to base’
  - that is, when the vectors are positioned so that they start at the same point
- We arrange this formula to make $\cos θ$ the subject:
- To find the angle between two vectors
  - Calculate the scalar product between them
  - Calculate the magnitude of each vector
  - Use the formula to find $\cos θ$
  - Use inverse trig to find $θ$

How do I find the angle between two lines?

To find the angle between two lines, find the angle between their direction vectors

- For example, if the lines have equations $r = a_{1} + s d_{1}$ and $r = a_{2} + t d_{2}$ , then the angle $θ$ between the lines is given by

$θ = \cos^{- 1} (\frac{d_{1} \cdot d_{2}}{|d_{1}| |d_{2}|})$

Worked example

Calculate the angle formed by the two vectors $v = (\binom{- 1}{\begin{matrix} 3 \\ 2 \end{matrix}})$ and $w = 3 i + 4 j - k$ .

al-fm-6-1-3-angle-between-lines-we-solution-png

Angle between Lines (Edexcel A Level Further Maths: Core Pure)

Revision Note

What is the scalar (dot) product?

What is the connection between the scalar product and trigonometry?

How do I tell if vectors or lines are perpendicular?

How do I find the angle between two vectors?

How do I find the angle between two lines?

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1. Complex Numbers

1.1 Complex Numbers & Argand Diagrams

1.1.1 Introduction to Complex Numbers

1.1.2 Solving Equations with Complex Roots

1.1.3 Modulus & Argument

1.1.4 Modulus-Argument Form

1.1.5 Loci in Argand Diagrams

1.1.6 Regions in Argand Diagrams

1.2 Exponential Form & de Moivre's Theorem

1.2.1 Exponential Form

1.2.2 de Moivre's Theorem

1.2.3 Applications of de Moivre's Theorem

1.2.4 Roots of Complex Numbers

2. Matrices

2.1 Properties of Matrices

2.1.1 Introduction to Matrices

2.1.2 Determinants of Matrices

2.1.3 Inverses of Matrices

2.2 Transformations using Matrices

2.2.1 Transformations using a Matrix

2.2.2 Geometric Transformations with Matrices

2.2.3 Invariant Points & Lines

3. Further Algebra & Functions

3.1 Roots of Polynomials

3.1.1 Roots of Polynomials

3.1.2 Linear Transformations of Roots

3.2 Series

3.2.1 Sums of Integers, Squares & Cubes

3.2.2 Method of Differences

3.3 Maclaurin Series

3.3.1 Maclaurin Series

4. Hyperbolic Functions

4.1 Hyperbolic Functions

4.1.1 Hyperbolic Functions & Graphs

4.1.2 Logarithmic Forms of Inverse Hyperbolic Functions

4.1.3 Hyperbolic Identities & Equations

4.1.4 Differentiating & Integrating Hyperbolic Functions

5. Further Calculus

5.1 Volumes of Revolution

5.1.1 Volumes of Revolution

5.1.2 Modelling with Volumes of Revolution

5.2 Methods in Calculus

5.2.1 Improper Integrals

5.2.2 Mean Value of a Function

5.2.3 Integrating with Partial Fractions

5.2.4 Calculus involving Inverse Trig

5.2.5 Integration by Substitution

6. Further Vectors

6.1 Vector Lines

6.1.1 Equations of Lines in 3D

6.1.2 Pairs of Lines in 3D

6.1.3 Angle between Lines

6.1.4 Shortest Distances - Lines

6.2 Vector Planes

6.2.1 Equations of planes

6.2.2 Combinations of Lines & Planes

6.2.3 Combinations of Planes

6.2.4 Shortest Distances - Planes

7. Polar Coordinates

7.1 Polar Coordinates

7.1.1 Polar Coordinates

7.1.2 Calculus with Polar Coordinates

8. Differential Equations

8.1 First Order Differential Equations

8.1.1 Intro to Differential Equations

8.1.2 Solving First Order Differential Equations

8.1.3 Modelling using First Order Differential Equations

8.2 Second Order Differential Equations

8.2.1 Solving Second Order Differential Equations

8.2.2 Coupled First Order Linear Equations