Vector Addition and Subtraction

When working with physical quantities like forces, velocities, or displacements, simply adding or subtracting their magnitudes isn't enough. These quantities have both magnitude and direction, making them vectors. To correctly combine or find the difference between two vectors, we need specific rules that account for their directional components. Understanding these operations is fundamental for fields ranging from physics and engineering to computer graphics and machine learning, where vectors represent data points or model parameters.

Geometric Vector Addition: The Head-to-Tail Rule

The head-to-tail rule provides an intuitive way to visualize vector addition. To add vector $\vec{A}$ and vector $\vec{B}$ , you place the tail of $\vec{B}$ at the head (arrow tip) of $\vec{A}$ . The resultant vector, $\vec{R} = \vec{A} + \vec{B}$ , is then drawn from the tail of $\vec{A}$ to the head of $\vec{B}$ . This method directly reflects how successive displacements or forces combine to produce a net effect.

Geometric Vector Addition (Head-to-Tail)

This plot shows two vectors,

\vec{A}

and

\vec{B}

, added using the head-to-tail rule. Vector

\vec{A}

starts at the origin. Vector

\vec{B}

begins where

\vec{A}

ends. The resultant vector

\vec{R}

connects the start of

\vec{A}

to the end of

\vec{B}

Loading chart...

Key Insight: The head-to-tail rule visually demonstrates how vectors combine sequentially, with the resultant representing the net change from start to finish.

💡 The Parallelogram Rule

Another geometric method, the parallelogram rule, is useful when vectors start from the same point. Draw $\vec{A}$ and $\vec{B}$ from a common origin. Complete the parallelogram using these two vectors as adjacent sides. The diagonal from the common origin to the opposite vertex is the resultant $\vec{A} + \vec{B}$ .

Algebraic Vector Addition: Component-Wise

While geometric methods are great for intuition, algebraic vector addition is how we perform calculations in practice. If vectors are represented by their components (e.g.,

\vec{A} = \begin{pmatrix} A_x \\ A_y \end{pmatrix}

and

\vec{B} = \begin{pmatrix} B_x \\ B_y \end{pmatrix}

), their sum is found by adding their corresponding components. This method is straightforward and scales easily to higher dimensions, making it the standard approach in computational tasks.

pythonAdding 2D Vectors Algebraically with NumPy

📐 Component-Wise Addition Formula

For two vectors

\vec{A} = \begin{pmatrix} A_x \\ A_y \end{pmatrix}

and

\vec{B} = \begin{pmatrix} B_x \\ B_y \end{pmatrix}

, their sum is:

\vec{A} + \vec{B} = \begin{pmatrix} A_x + B_x \\ A_y + B_y \end{pmatrix}

Properties of Vector Addition

Vector addition shares several properties with scalar addition. It is commutative, meaning the order of addition does not affect the result ( $\vec{A} + \vec{B} = \vec{B} + \vec{A}$ ). This is evident in both the head-to-tail and parallelogram rules. Vector addition is also associative, allowing us to group multiple vectors in any order when adding them ( $\vec{A} + (\vec{B} + \vec{C}) = (\vec{A} + \vec{B}) + \vec{C}$ ). These properties simplify complex vector calculations.

Check Your Understanding

If you add two displacement vectors, what does the resultant vector represent?

Geometric Vector Subtraction: Adding the Negative Vector

Vector subtraction, $\vec{A} - \vec{B}$ , can be thought of as adding the negative of vector $\vec{B}$ to vector $\vec{A}$ . The negative of a vector, $-\vec{B}$ , has the same magnitude as $\vec{B}$ but points in the exact opposite direction. Geometrically, you draw $\vec{A}$ and then place the tail of $-\vec{B}$ at the head of $\vec{A}$ . The resultant is drawn from the tail of $\vec{A}$ to the head of $-\vec{B}$ .

Geometric Vector Subtraction (A - B = A + (-B))

This plot illustrates vector subtraction by showing

\vec{A}

, the negative vector

-\vec{B}

, and their head-to-tail sum which represents

\vec{A} - \vec{B}

. Vector

\vec{B}

is shown for reference, and

-\vec{B}

points in the opposite direction.

Loading chart...

Key Insight: Vector subtraction is geometrically equivalent to adding the reversed version of the second vector, providing a clear visual for finding the difference.

Algebraic Vector Subtraction: Component-Wise

Similar to addition, algebraic vector subtraction involves subtracting the corresponding components of the vectors. If

\vec{A} = \begin{pmatrix} A_x \\ A_y \end{pmatrix}

and

\vec{B} = \begin{pmatrix} B_x \\ B_y \end{pmatrix}

, then

\vec{A} - \vec{B}

is found by subtracting

B_x

from

A_x

and

B_y

from

A_y

. This component-wise approach is efficient and directly implements the concept of adding a negative vector, as subtracting a positive number is the same as adding a negative number.

pythonSubtracting 2D Vectors Algebraically with NumPy

⚠️ Order Matters for Subtraction

Unlike addition, vector subtraction is not commutative. $\vec{A} - \vec{B}$ is generally not equal to $\vec{B} - \vec{A}$ . In fact, $\vec{A} - \vec{B} = -(\vec{B} - \vec{A})$ . This means the order in which you subtract vectors is critical to obtaining the correct resultant.

Vectors in Higher Dimensions

The principles of component-wise addition and subtraction extend seamlessly to vectors in three or more dimensions. For a 3D vector, you simply add or subtract the $z$ -components along with the $x$ and $y$ components. This scalability is why algebraic methods are so powerful in fields like machine learning, where data points often exist in hundreds or thousands of dimensions. The underlying operation remains the same: combine corresponding elements.

pythonAdding 3D Vectors for Object Position

Try It Yourself

Given two 3D force vectors, F1 = [15, -8, 12] and F2 = [5, 10, -3], calculate their resultant force and the difference F1 - F2. Print both results.

python

Check Your Understanding

Which operation is NOT commutative for vectors?

Comparing Vector Addition and Subtraction

This plot shows two base vectors,

\vec{A}

and

\vec{B}

, originating from the origin. It then illustrates their sum,

\vec{A} + \vec{B}

, and their difference,

\vec{A} - \vec{B}

, both also originating from the origin. This highlights how both operations yield new vectors.

Loading chart...

Key Insight: Vector addition and subtraction both produce new vectors, but their geometric and algebraic results are distinct, reflecting different combinations of direction and magnitude.

Key Takeaways

Vector addition combines two vectors to find a resultant vector, representing a net effect or sum of quantities with direction.
Geometrically, vector addition follows the head-to-tail rule (or parallelogram rule), where the resultant connects the start of the first vector to the end of the last.
Algebraically, vector addition is performed component-wise, adding corresponding components (e.g., $A_x + B_x$ , $A_y + B_y$ ).
Vector addition is commutative and associative, meaning the order of addition and grouping does not change the result.
Vector subtraction, $\vec{A} - \vec{B}$ , is equivalent to adding the negative of the second vector, $\vec{A} + (-\vec{B})$ .
Algebraic vector subtraction is also component-wise, subtracting corresponding components (e.g., $A_x - B_x$ , $A_y - B_y$ ).
Vector subtraction is not commutative; the order of subtraction significantly impacts the resultant vector.

← All lessons in Linear Algebra: Vectors

Vector Addition and Subtraction

Geometric Vector Addition: The Head-to-Tail Rule

Algebraic Vector Addition: Component-Wise

Properties of Vector Addition

Geometric Vector Subtraction: Adding the Negative Vector

Algebraic Vector Subtraction: Component-Wise

Vectors in Higher Dimensions

Ready to keep this from fading?