Edexcel IAL - Mechanics 1- 2.1 Magnitude and Direction of Vectors- Study notes  - New syllabus

Magnitude, Direction, and Resultant of Vectors

A vector represents a quantity that has both magnitude and direction. In mechanics, vectors are commonly used to represent displacement, velocity, acceleration, and force.

Magnitude of a Vector

If a vector is written in component form as \( \langle a, b \rangle \) or \( a\mathbf{i} + b\mathbf{j} \), its magnitude is given by

\( |\mathbf{v}| = \sqrt{a^2 + b^2} \)

This formula follows directly from the Pythagorean Theorem.

Direction of a Vector

The direction of a vector is often described by the angle \( \theta \) it makes with the positive \( x \)-axis. For a vector \( \langle a, b \rangle \),

\( \tan \theta = \dfrac{b}{a} \)

The correct quadrant must always be considered when determining the direction.

Resultant of Vectors

The resultant of two or more vectors is the single vector that has the same effect as all the vectors acting together.

If

\( \mathbf{u} = \langle a_1, b_1 \rangle \), \( \mathbf{v} = \langle a_2, b_2 \rangle \),

then the resultant is

\( \mathbf{R} = \mathbf{u} + \mathbf{v} = \langle a_1 + a_2,\; b_1 + b_2 \rangle \)

Graphically, the resultant is found using a vector diagram by placing vectors head to tail.

Resolving a Vector into Components

A vector of magnitude \( r \) making an angle \( \theta \) with the positive \( x \)-axis can be resolved into perpendicular components:

Horizontal component: \( r\cos \theta \)

Vertical component: \( r\sin \theta \)

So the vector can be written as

\( r\cos \theta\,\mathbf{i} + r\sin \theta\,\mathbf{j} \)

Unit Vector Form

Using unit vectors \( \mathbf{i} = \langle 1,0 \rangle \) and \( \mathbf{j} = \langle 0,1 \rangle \), any vector in \( \mathbb{R}^2 \) can be expressed as

\( \mathbf{v} = a\mathbf{i} + b\mathbf{j} \)