2.1 Basics of vectors

Achievable AP Physics 1

2. Vectors and their analysis

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Basics of vectors

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Vectors are essential in AP Physics 1 because they let you describe quantities that have both magnitude and direction. Here’s what this section focuses on:

Fundamentals of vectors
Graphical representation of vectors
Methods for addition and subtraction.

(Note: A more detailed look at component analysis will be covered in the next sub-chapter.)

What is a vector?

A vector is a quantity that has both magnitude and direction. This contrasts with a scalar, which has only magnitude (such as mass or temperature).

Notation: Vectors are typically written in bold (e.g., v) or with an arrow over the letter (e.g., $v$ ). The magnitude of a vector $v$ is written as $∣ v ∣$ .

Representing vectors on the coordinate plane

We usually draw a vector as an arrow:

The length of the arrow represents the magnitude.
The direction of the arrow shows which way the vector points.

In two dimensions, a vector can be written in component form as:

$v = v_{x} \hat{i} + v_{y} \hat{j},$

where:

$v_{x}$ is the component along the $x$ -axis,
$v_{y}$ is the component along the $y$ -axis,
$\hat{i}$ and $\hat{j}$ are the unit vectors in the $x$ and $y$ directions respectively.

For example, the vector with components $v_{x} = 3$ and $v_{y} = 4$ can be written in component form as

$v = 3 \hat{i} + 4 \hat{j},$

or equivalently in coordinate form as $(3, 4)$ .

Finding the magnitude of a vector

The magnitude (or length) of a vector tells you how large the vector is, independent of its direction. For a vector written in component form:

$v = v_{x} \hat{i} + v_{y} \hat{j},$

you find the magnitude using the Pythagorean theorem:

$∣ v ∣ = v_{x}^{2} + v_{y}^{2} .$

Example: If $v = (3, 4)$ , then the magnitude is:

$∣ v ∣ = 3^{2} + 4^{2} = 9 + 16 = 25 = 5.$

Being able to compute magnitude matters because many physics vectors - like displacement, velocity, and force - often need to be reported as a single “size” value.

Vector addition and subtraction methods

In physics, you often need to combine vectors to find a net effect (for example, the net force on an object). Two common graphical methods are the tip-to-tail method and the parallelogram method.

Tip-to-tail method

Procedure:

Draw the first vector with its tail at a chosen starting point.
Place the tail of the second vector at the tip of the first.
For additional vectors, continue by placing each new vector’s tail at the tip of the previous one.
Draw the resultant vector from the tail of the first vector to the tip of the final vector.

This method forms a chain of vectors, making it easy to see how directions and magnitudes combine.

Parallelogram method

Procedure:

Draw the two vectors so that their tails start at the same point.
Construct a parallelogram using the two vectors as adjacent sides.
Draw the diagonal from the common tail; that diagonal is the resultant vector.

The parallelogram method highlights the geometry of vector addition.
It’s useful for quickly estimating the magnitude and direction of the resultant vector.

Subtraction of vectors

Vector subtraction is done by adding the negative of a vector. To compute $A - B$ :

Reverse the direction of $B$ to get $- B$ .
Add $A$ and $- B$ using either the tip-to-tail or parallelogram method.

For example, if

$A = (A_{x}, A_{y}) and B = (B_{x}, B_{y}),$

then:

$A - B = (A_{x} - B_{x}, A_{y} - B_{y}) .$

Example problem 1

Given: Vector A = $(3, 4)$ Vector B = $(- 2, 5)$

Find: $A + B$

Solution:

(spoiler)

Components: $A_{x} = 3, A_{y} = 4$ $B_{x} = - 2, B_{y} = 5$
Sum components: $(A + B)_{x} = 3 + (- 2) = 1$ $(A + B)_{y} = 4 + 5 = 9$
Resultant: $A + B = (1, 9)$
Interpretation: The resultant vector points 1 unit to the right (positive x-direction) and 9 units upward (positive y-direction).

Example problem 2

Given: Vector E = $(2, 1)$ Vector F = $(- 1, 3)$ Vector G = $(4, - 2)$

Find: $E + F + G$

Solution:

(spoiler)

Components:

$E_{x} = 2, E_{y} = 1$ $F_{x} = - 1, F_{y} = 3$ $G_{x} = 4, G_{y} = - 2$
Sum components:

$(E + F + G)_{x} = 2 + (- 1) + 4 = 5$ $(E + F + G)_{y} = 1 + 3 + (- 2) = 2$
Resultant:

$E + F + G = (5, 2)$
Interpretation: The resultant vector points 5 units to the right and 2 units upward, representing the combined effect of all three vectors.

Fundamentals of vectors

Vectors: quantities with both magnitude and direction
Scalars: quantities with only magnitude
Notation: boldface ( v) or arrow ( $v$ ); magnitude as $∣ v ∣$ Graphical representation of vectors
Drawn as arrows: length = magnitude, direction = vector direction
Component form: $v = v_{x} \hat{i} + v_{y} \hat{j}$
- $v_{x}$ , $v_{y}$ : components along $x$ and $y$ axes
- $\hat{i}$ , $\hat{j}$ : unit vectors in $x$ , $y$ directions

Finding the magnitude of a vector

Formula: $∣ v ∣ = v_{x}^{2} + v_{y}^{2}$
Use Pythagorean theorem for 2D vectors
Magnitude represents the “size” of the vector

Vector addition and subtraction methods

Tip-to-tail method:
- Place each new vector’s tail at the previous vector’s tip
- Resultant: from first tail to last tip
Parallelogram method:
- Draw vectors from common origin
- Resultant: diagonal of parallelogram from common tail
Subtraction:
- Reverse direction of vector to be subtracted
- Add using tip-to-tail or parallelogram method
- Component subtraction: $(A_{x} - B_{x}, A_{y} - B_{y})$

Worked examples

Add vectors by summing $x$ and $y$ components separately
Resultant vector: new components represent combined effect
Interpret direction and magnitude from resultant components

Next | 2.2 Components analysis

Basics of vectors

Vectors are essential in AP Physics 1 because they let you describe quantities that have both magnitude and direction. Here’s what this section focuses on:

Fundamentals of vectors
Graphical representation of vectors
Methods for addition and subtraction.

(Note: A more detailed look at component analysis will be covered in the next sub-chapter.)

What is a vector?

A vector is a quantity that has both magnitude and direction. This contrasts with a scalar, which has only magnitude (such as mass or temperature).

Notation: Vectors are typically written in bold (e.g., v) or with an arrow over the letter (e.g., $v$ ). The magnitude of a vector $v$ is written as $∣ v ∣$ .

Representing vectors on the coordinate plane

We usually draw a vector as an arrow:

The length of the arrow represents the magnitude.
The direction of the arrow shows which way the vector points.

In two dimensions, a vector can be written in component form as:

$v = v_{x} \hat{i} + v_{y} \hat{j},$

where:

$v_{x}$ is the component along the $x$ -axis,
$v_{y}$ is the component along the $y$ -axis,
$\hat{i}$ and $\hat{j}$ are the unit vectors in the $x$ and $y$ directions respectively.

For example, the vector with components $v_{x} = 3$ and $v_{y} = 4$ can be written in component form as

$v = 3 \hat{i} + 4 \hat{j},$

or equivalently in coordinate form as $(3, 4)$ .

Finding the magnitude of a vector

The magnitude (or length) of a vector tells you how large the vector is, independent of its direction. For a vector written in component form:

$v = v_{x} \hat{i} + v_{y} \hat{j},$

you find the magnitude using the Pythagorean theorem:

$∣ v ∣ = v_{x}^{2} + v_{y}^{2} .$

Example: If $v = (3, 4)$ , then the magnitude is:

$∣ v ∣ = 3^{2} + 4^{2} = 9 + 16 = 25 = 5.$

Being able to compute magnitude matters because many physics vectors - like displacement, velocity, and force - often need to be reported as a single “size” value.