1.2.3. Vector Addition and Subtraction

Table of Contents

Combining Vectors

Vector addition and subtraction are the basic operations used to combine physical quantities that have both size and direction. When two vectors act together, we often want to know their single combined effect. This is called the resultant vector.

If $\vec{A}$ and $\vec{B}$ are two vectors, their sum is written as

$$
\vec{R} = \vec{A} + \vec{B}
$$

The vector $\vec{R}$ is called the resultant.

Vector addition is not the same as ordinary number addition, because direction matters. A vector pointing east and a vector pointing north do not simply become a larger number. They combine into a new vector with its own magnitude and direction.

When adding vectors, you must combine both magnitude and direction. You cannot add vector magnitudes alone unless the vectors lie along the same line and point in the same or opposite directions.

Head-to-Tail Method

A simple geometric way to add vectors is the head-to-tail method. Draw the first vector. Then place the tail of the second vector at the head of the first. The resultant vector goes from the tail of the first vector to the head of the second.

If a person walks $3 \, \text{m}$ east and then $4 \, \text{m}$ north, the displacement is not $7 \, \text{m}$. Instead, the resultant is the diagonal of a right triangle.

Rendering scientific sketch…

Head-to-tail vector addition

This method works for any number of vectors. You place each new vector at the end of the previous one, and then draw the resultant from the starting point to the final point.

Parallelogram Method

Another geometric method for adding two vectors is the parallelogram method. Draw both vectors starting from the same point. Then complete a parallelogram. The diagonal from the common starting point gives the sum.

Rendering scientific sketch…

Parallelogram method

This method gives the same result as the head-to-tail method. It is especially useful when visualizing two forces acting at the same point.

Vector Subtraction

Subtracting a vector means adding its opposite. The opposite of $\vec{B}$ is written $-\vec{B}$. It has the same magnitude as $\vec{B}$ but points in the opposite direction.

So,

$$
\vec{A} - \vec{B} = \vec{A} + (-\vec{B})
$$

To subtract vectors geometrically, reverse the direction of the vector being subtracted, then use vector addition.

Rendering scientific sketch…

Vector subtraction as addition of the opposite

This is important in physics. For example, a change in position is found by subtracting one position vector from another.

Vector subtraction rule:
$$
\vec{A} - \vec{B} = \vec{A} + (-\vec{B})
$$
Always reverse the direction of the vector being subtracted before combining.

Addition in One Dimension

When vectors lie on the same straight line, addition and subtraction become simpler. You can assign one direction as positive and the opposite direction as negative.

For example, let east be positive. Then a displacement of $5 \, \text{m}$ east is $+5 \, \text{m}$, and a displacement of $2 \, \text{m}$ west is $-2 \, \text{m}$.

Their sum is

$$
(+5) + (-2) = +3 \, \text{m}
$$

So the resultant is $3 \, \text{m}$ east.

This shows that in one dimension, vector addition can be treated like signed number addition.

Properties of Vector Addition

Vector addition follows several useful rules.

The commutative property means that the order does not matter:

$$
\vec{A} + \vec{B} = \vec{B} + \vec{A}
$$

The associative property means that grouping does not matter:

$$
(\vec{A} + \vec{B}) + \vec{C} = \vec{A} + (\vec{B} + \vec{C})
$$

There is also a zero vector, written $\vec{0}$, which has zero magnitude and no specific direction:

$$
\vec{A} + \vec{0} = \vec{A}
$$

And every vector has an opposite:

$$
\vec{A} + (-\vec{A}) = \vec{0}
$$

These properties make vector algebra consistent and predictable.

Important vector properties:
$$
\vec{A} + \vec{B} = \vec{B} + \vec{A}
$$
$$
(\vec{A} + \vec{B}) + \vec{C} = \vec{A} + (\vec{B} + \vec{C})
$$
$$
\vec{A} - \vec{B} = \vec{A} + (-\vec{B})
$$

Resultant Magnitude for Perpendicular Vectors

When two vectors are perpendicular, the magnitude of their sum can be found using the Pythagorean theorem.

If $\vec{A}$ and $\vec{B}$ are at right angles, then

$$
|\vec{A} + \vec{B}| = \sqrt{A^2 + B^2}
$$

where $A = |\vec{A}|$ and $B = |\vec{B}|$.

For example, if one vector is $3$ units east and the other is $4$ units north, then the resultant magnitude is

$$
\sqrt{3^2 + 4^2} = 5
$$

So the resultant has magnitude $5$ units.

Its direction must also be specified. In this example, it points northeast.

Common Physical Examples

Vector addition appears everywhere in physics. Two forces acting on an object combine by vector addition to produce a net force. Two displacements combine to give a total displacement. Two velocities may combine to give a resultant velocity.

The idea is always the same. Each vector keeps its magnitude and direction, and the final result depends on both.

The following table shows some common interpretations.

Vector operation	Physical meaning
$\vec{F}_1 + \vec{F}_2$	Net force
$\vec{d}_1 + \vec{d}_2$	Total displacement
$\vec{v}_1 + \vec{v}_2$	Resultant velocity
$\vec{r}_2 - \vec{r}_1$	Change in position

A Simple Example

Suppose a box is pulled by one force of $6 \, \text{N}$ to the right and another force of $2 \, \text{N}$ to the left. Since both vectors lie on the same line, take right as positive:

$$
\vec{F}_{\text{net}} = +6 + (-2) = +4 \, \text{N}
$$

So the net force is $4 \, \text{N}$ to the right.

Now suppose instead the forces are $6 \, \text{N}$ east and $8 \, \text{N}$ north. Then they are perpendicular, so the resultant magnitude is

$$
|\vec{F}_{\text{net}}| = \sqrt{6^2 + 8^2} = 10 \, \text{N}
$$

The net force is not east or north alone. It points in a direction between them.

Final Idea

Vector addition and subtraction let us combine directional quantities into a single equivalent vector. Addition can be done geometrically by head-to-tail placement or by the parallelogram method. Subtraction is handled by adding the opposite vector. These operations are fundamental in physics because many important quantities, such as force, displacement, and velocity, combine in this way.

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1.2.3. Vector Addition and Subtraction

Combining Vectors

Head-to-Tail Method

Parallelogram Method

Vector Subtraction

Addition in One Dimension

Properties of Vector Addition

Resultant Magnitude for Perpendicular Vectors

Common Physical Examples

A Simple Example

Final Idea

Comments

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