Magnitude, dot product and angle between vectors

Lengths, dot/cross products and the angle between vectors without coordinate algebra.

In short: The magnitude of a vector is its length, the dot product multiplies matching components and adds them, and the angle between two vectors comes from the dot product divided by the product of their magnitudes. The GDC computes these directly from vectors entered as lists or matrices. Use inverse cosine to get the angle.

When you'd use this

Finding the length (magnitude) of a vector without working through Pythagoras by hand.
Finding the angle between two vectors using the dot product formula.
Checking whether two vectors are perpendicular (dot product = 0).
A 3D vector question where the arithmetic would be slow and error-prone by hand.

TI-84 Plus CE Casio fx-CG50 and fx-CG100 TI-Nspire CX

At a glance: TI-84 Plus CE, Casio fx-CG50 and TI-Nspire CX compared

	TI-84 Plus CE	Casio fx-CG50 and fx-CG100	TI-Nspire CX
Key sequence	Store components in lists; magnitude = √(sum of squares); dot product = sum(L1×L2).	Run-Matrix → MAT/VCT to enter vectors; OPTN gives DotP, CrossP and the norm (magnitude).	Type vectors with the matrix template; use dotP(), crossP() and norm() from menu → Matrix & Vector → Vector.

On a TI-84 Plus CE

Enter each vector's components (as a list or a 3×1 matrix).
Store components in lists; magnitude = √(sum of squares); dot product = sum(L1×L2).
Angle between: cos θ = (a·b)/(|a||b|), then inverse cosine.

Tip: A dot product of 0 means the vectors are perpendicular - the quickest perpendicularity check.

On a Casio fx-CG50 and fx-CG100

Enter each vector's components (as a list or a 3×1 matrix).
Run-Matrix → MAT/VCT to enter vectors; OPTN gives DotP, CrossP and the norm (magnitude).
Angle between: cos θ = (a·b)/(|a||b|), then inverse cosine.

Tip: A dot product of 0 means the vectors are perpendicular - the quickest perpendicularity check.

On a TI-Nspire CX

Enter each vector's components (as a list or a 3×1 matrix).
Type vectors with the matrix template; use dotP(), crossP() and norm() from menu → Matrix & Vector → Vector.
Angle between: cos θ = (a·b)/(|a||b|), then inverse cosine.

Tip: A dot product of 0 means the vectors are perpendicular - the quickest perpendicularity check.

Try it yourself

Here's a real IB-style question that uses exactly this technique.

Medium Calculator Paper 2 [3 marks]

For a = (3, 4, 0) and b = (1, 2, 2), find a · b and the angle between a and b, to 3 significant figures.

a · b = 11, angle = 42.8°

Mark it

Correct 3 / 3 marks

Worked solution & mark scheme:

M1 a · b = 3(1) + 4(2) + 0(2) = 11

M1 cosθ = 11 ÷ (|a||b|)

A1 θ = 42.8°

Common questions

When would I need to find the magnitude, dot product and angle between vectors in IB Maths?

Lengths, dot/cross products and the angle between vectors without coordinate algebra. Finding the length (magnitude) of a vector without working through Pythagoras by hand. Finding the angle between two vectors using the dot product formula.

How do I find the magnitude, dot product and angle between vectors on a TI-84 Plus CE?

1. Enter each vector's components (as a list or a 3×1 matrix). 2. Store components in lists; magnitude = √(sum of squares); dot product = sum(L1×L2). 3. Angle between: cos θ = (a·b)/(|a||b|), then inverse cosine.

How do I find the magnitude, dot product and angle between vectors on a Casio fx-CG50 and fx-CG100?

1. Enter each vector's components (as a list or a 3×1 matrix). 2. Run-Matrix → MAT/VCT to enter vectors; OPTN gives DotP, CrossP and the norm (magnitude). 3. Angle between: cos θ = (a·b)/(|a||b|), then inverse cosine.

How do I find the magnitude, dot product and angle between vectors on a TI-Nspire CX?

1. Enter each vector's components (as a list or a 3×1 matrix). 2. Type vectors with the matrix template; use dotP(), crossP() and norm() from menu → Matrix & Vector → Vector. 3. Angle between: cos θ = (a·b)/(|a||b|), then inverse cosine.

What should I watch out for when I find the magnitude, dot product and angle between vectors?

A dot product of 0 means the vectors are perpendicular - the quickest perpendicularity check.

How are marks awarded when I find the magnitude, dot product and angle between vectors in an IB exam?

In the worked example on this page (3 marks, Paper 2), the marks are: M1: a · b = 3(1) + 4(2) + 0(2) = 11; M1: cosθ = 11 ÷ (|a||b|); A1: θ = 42.8°.

Practise with your calculator

Questions that need this technique link back to this guide. Try one in practice mode, or see every GDC guide.