Vector Basics (AI HL)

Q: How do you find the magnitude of a vector?

For a vector v with components x, y (and z in 3D), the magnitude is |v| = sqrt(x^2 + y^2 + z^2) - the vector's length, found using Pythagoras' theorem extended to three dimensions.

Q: What is a unit vector?

A unit vector has magnitude 1. To rescale any vector v into a unit vector in the same direction, divide every component by |v|.

Before angles or lines come into it, a vector question usually starts with two things: how long is this vector, and what does it look like scaled down to length \(1\)? This page covers magnitude and unit vectors, plus the resultant of two or more vectors added together, with worked examples and the mistakes that lose marks. It's part of the broader Vectors topic.

12 questions on this sub-topic.

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Magnitude and unit vectors

Covered under IB syllabus reference AHL3.10: the concept of a vector and a scalar, the magnitude of a vector from its components, and rescaling a vector into a unit vector.

Magnitude of a vector

\[|\mathbf v| = \sqrt{x^2+y^2+z^2}\]

The vector's length - Pythagoras' theorem extended to three components. Drop the \(z^2\) term for a 2D vector.

✓ In the formula booklet

Unit vector

\[\hat{\mathbf v} = \dfrac{1}{|\mathbf v|}\mathbf v\]

Divide every component by the magnitude to rescale a vector to length \(1\) without changing its direction.

✓ In the formula booklet

Need the full syllabus wording, base vectors \(\mathbf i,\mathbf j,\mathbf k\), or GDC vector tools? See Vectors.

Worked examples

Easy

GDC

[4 marks]

Given \(\mathbf v = (3, -4)\):

(a) Find \(|\mathbf v|.\)
(b) Find the unit vector in the direction of \(\mathbf v.\)

Worked solution

(a) \(|\mathbf v| = \sqrt{9 + 16}\) M1
\(= 5.\) A1

(b) \(\hat{\mathbf v} = \tfrac15(3, -4)\) M1
\(= (0.6, -0.8).\) A1

A GDC is permitted on this paper, so you may evaluate or verify this result directly on the calculator.

M1 \(\sqrt{x^2+y^2}\) A1 Correct answer of \(5\) M1 \(\tfrac{\mathbf v}{|\mathbf v|}\) A1 Unit vector

Medium

GDC

[4 marks]

A drone flies from \(P(0,0,0)\) to \(Q(40, 30, 20)\) (m).

(a) Find the displacement vector and its magnitude.
(b) If the flight takes 10 s at constant velocity, find the velocity vector and speed.

Worked solution

(a) \(\vec{PQ} = (40, 30, 20);\ |\vec{PQ}| = \sqrt{2900}\) M1
\(\approx 53.9\) m. A1

(b) \(\mathbf v = \tfrac{1}{10}(40, 30, 20) = (4, 3, 2)\) m/s; speed \(= \sqrt{29}\) M1
\(\approx 5.39\) m/s. A1

M1 Displacement & magnitude A1 Correct answer of \(\approx53.9\) M1 \(\tfrac{\text{displacement}}{t}\) A1 Speed \(\approx5.39\)

Common mistakes

Treating magnitude as a vector. \(|\mathbf v|\) is a single number (a scalar) - don't write it in component form or try to add it directly to another vector.
Dividing only one component when finding a unit vector. Every component must be divided by the same magnitude \(|\mathbf v|\), not just the first one - a partial rescale changes the vector's direction as well as its length.
Rounding the magnitude too early. If \(|\mathbf v|\) is used again in the same question (for a unit vector, a speed, or a further calculation), keep it exact or unrounded until the final answer - early rounding compounds into an inaccurate final value.

Full topicVectors RelatedVector equations of lines RelatedDot product and angles FormulasAI HL formula booklet explained PrintAI HL formula sheet

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Quick answers

How do you find the magnitude of a vector?

For a vector \(\mathbf v\) with components \(x, y\) (and \(z\) in 3D), the magnitude is \(|\mathbf v| = \sqrt{x^2+y^2+z^2}\) - the vector's length, found using Pythagoras' theorem extended to three dimensions.

What is a unit vector?

A unit vector has magnitude \(1\). To rescale any vector \(\mathbf v\) into a unit vector in the same direction, divide every component by \(|\mathbf v|\).

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