The Scalar Product | DP IB Maths: AA HL Revision Notes 2021

The Scalar ('Dot') Product

What is the scalar product?

The scalar product (also known as the dot product) is one form in which two vectors can be combined together
The scalar product between two vectors a and b is denoted $a \cdot b$
The result of taking the scalar product of two vectors is a real number

i.e. a scalar

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)

How is the scalar product calculated?

There are two methods for calculating the scalar product
The most common method used to find the scalar product between the two vectors v and w is to find the sum of the product of each component in the two vectors

$v \cdot w = v_{1} w_{1} + v_{2} w_{2} + v_{3} w_{3}$
Where $v = (\binom{v_{1}}{\begin{matrix} v_{2} \\ v_{3} \end{matrix}})$ and $w = (\binom{w_{1}}{\begin{matrix} w_{2} \\ w_{3} \end{matrix}})$
This is given in the formula booklet

The scalar product is also equal to the product of the magnitudes of the two vectors and the cosine of the angle between them

$v \cdot w = |v| |w| \cos θ$
Where θ is the angle between v and w

The two vectors v and w are joined at the start and pointing away from each other

The scalar product can be used in the second formula to find the angle between the two vectors

What properties of the scalar product do I need to know?

The order of the vectors doesn’t change the result of the scalar product (it is commutative)
- $v \cdot w = w \cdot v$
The distributive law over addition can be used to ‘expand brackets’
- $u \cdot (v + w) = u \cdot v + u \cdot w$
The scalar product is associative with respect to multiplication by a scalar
- $(k v) \cdot (w) = k (v \cdot w)$
The scalar product between a vector and itself is equal to the square of its magnitude
- $v \cdot v = {|v|}^{2}$
If two vectors, v and w, are parallel then the magnitude of the scalar product is equal to the product of the magnitudes of the vectors
- $| v \cdot w | = | w | | v |$
- This is because cos 0° = 1 and cos 180° = -1
If two vectors are perpendicular the scalar product is zero
- This is because cos 90° = 0

Examiner Tip

Whilst the formulae for the scalar product are given in the formula booklet, the properties of the scalar product are not, however they are important and it is likely that you will need to recall them in your exam so be sure to commit them to memory

Worked example

Calculate the scalar product between the two vectors $v = (\binom{2}{\begin{matrix} 0 \\ - 5 \end{matrix}})$ and $w = 3 j - 2 k - i$ using:

the formula

v \cdot w = v_{1} w_{1} + v_{2} w_{2} + v_{3} w_{3}

3-9-4-ib-aa-hl-the-scalar-product-we-solution-a

ii)

the formula

v \cdot w = |v| |w| \cos θ

, given that the angle between the two vectors is 66.6°.

3-9-4-ib-aa-hl-the-scalar-product-we-solution-b

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Angle Between Two Vectors

How do I find the angle between two vectors?

If two vectors with different directions are placed at the same starting position, they will form an angle between them
The two formulae for the scalar product can be used together to find this angle

$\cos θ = \frac{v_{1} w_{1} + v_{2} w_{2} + v_{3} w_{3}}{|v| | w |}$
This is given in the formula booklet

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

Examiner Tip

The formula for this is given in the formula booklet so you do not need to remember it but make sure that you can find it quickly and easily in your exam

Worked example

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

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

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Perpendicular Vectors

How do I know if two vectors are perpendicular?

If the scalar product of two (non-zero) vectors is zero then they are perpendicular

If $v \cdot w = 0$ then v and w must be perpendicular to each other

Two vectors are perpendicular if their scalar product is zero

The value of cos θ = 0 therefore |v||w|cos θ = 0

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

The Scalar Product (DP IB Maths: AA HL)

Revision Note

The Scalar ('Dot') Product

What is the scalar product?

How is the scalar product calculated?

What properties of the scalar product do I need to know?

Examiner Tip

Worked example

Angle Between Two Vectors

How do I find the angle between two vectors?

Examiner Tip

Worked example

Perpendicular Vectors

How do I know if two vectors are perpendicular?

Worked example

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Author: Amber