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Lesson 3: Magnitude, Unit Vectors & Dot Product Intro

1. Magnitude (Length) of a Vector

The magnitude (or norm) of a vector is its length, calculated using the Pythagorean theorem in higher dimensions.

Formula (2D): \(\|\vec{v}\| = \sqrt{x^2 + y^2}\)

Example: \(\|\begin{pmatrix} 3 \\ 4 \end{pmatrix}\| = \sqrt{9 + 16} = \sqrt{25} = 5\)

Exercise 1

Find the magnitude:
\[ \left\| \begin{pmatrix} 5 \\ 12 \end{pmatrix} \right\| = \sqrt{ x + y } = z \] x =
y =
z =

2. Unit Vector

A unit vector has magnitude 1. It shows only direction.

To get unit vector: divide original vector by its magnitude.

Example: unit vector of \(\begin{pmatrix} 3 \\ 4 \end{pmatrix}\) is \(\begin{pmatrix} 3/5 \\ 4/5 \end{pmatrix}\)

Exercise 2

What is the unit vector in the direction of \(\begin{pmatrix} 8 \\ 6 \end{pmatrix}\)?

3. Introduction to Dot Product

The dot product of two vectors is a number (scalar) that tells us how aligned they are.

Formula (2D): \(\vec{a} \cdot \vec{b} = a_x b_x + a_y b_y\)

Example: \(\begin{pmatrix} 1 \\ 2 \end{pmatrix} \cdot \begin{pmatrix} 3 \\ 4 \end{pmatrix} = 1·3 + 2·4 = 11\)

If dot product = 0 → vectors are perpendicular.

Exercise 3

Compute the dot product:
\[ \begin{pmatrix} 2 \\ 5 \end{pmatrix} \cdot \begin{pmatrix} 3 \\ -1 \end{pmatrix} = x \cdot 3 + y \cdot (-1) = z \] x =
y =
z =

Summary – Lesson 3

Next lessons: more on dot product, cross product (3D), and projections.

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