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Vector and Operations

Vector Subtraction

Nabil Akbarazzima Fatih

Mathematics

Basic Concept of Vector Subtraction

Vector subtraction is one of the basic operations in vector mathematics. Unlike vector addition that combines two vectors, vector subtraction produces the difference between two vectors. When we subtract vector b⃗\vec{b} from vector a⃗\vec{a}, the result is a vector that, when added to b⃗\vec{b}, will produce a⃗\vec{a}.

Mathematically, vector subtraction is defined as:

a⃗−b⃗=a⃗+(−b⃗)\vec{a} - \vec{b} = \vec{a} + (-\vec{b})

This means that subtracting vector b⃗\vec{b} from vector a⃗\vec{a} is equivalent to adding vector a⃗\vec{a} with the negative of vector b⃗\vec{b}.

Geometric Vector Subtraction

Geometrically, vector subtraction a⃗−b⃗\vec{a} - \vec{b} can be depicted by:

  1. Drawing vectors a⃗\vec{a} and b⃗\vec{b} with the same initial point.
  2. Drawing vector −b⃗-\vec{b} (vector b⃗\vec{b} with reversed direction).
  3. Drawing a vector from the endpoint of b⃗\vec{b} to the endpoint of a⃗\vec{a}.

The resulting vector, a⃗−b⃗\vec{a} - \vec{b}, can also be obtained by drawing a line from the endpoint of vector b⃗\vec{b} to the endpoint of vector a⃗\vec{a} when both vectors are drawn from the same origin point.

Visualization of Vector Subtraction in 3D Space
Vector subtraction a - b (purple color) is equivalent to vector a (blue color) added to the negative of vector b (red color).

Algebraic Vector Subtraction

Vector subtraction can be performed by subtracting corresponding components. Suppose we have two vectors:

a⃗=(a1,a2,a3)\vec{a} = (a_1, a_2, a_3)
b⃗=(b1,b2,b3)\vec{b} = (b_1, b_2, b_3)

Then vector subtraction a⃗−b⃗\vec{a} - \vec{b} can be calculated as:

a⃗−b⃗=(a1−b1,a2−b2,a3−b3)\vec{a} - \vec{b} = (a_1 - b_1, a_2 - b_2, a_3 - b_3)

For two-dimensional vectors, the equation becomes:

a⃗−b⃗=(a1−b1,a2−b2)\vec{a} - \vec{b} = (a_1 - b_1, a_2 - b_2)

Example of Vector Subtraction Calculation

Suppose there are two vectors:

a⃗=(3,4)\vec{a} = (3, 4)
b⃗=(1,2)\vec{b} = (1, 2)

Vector subtraction a⃗−b⃗\vec{a} - \vec{b} is:

a⃗−b⃗=(3,4)−(1,2)=(3−1,4−2)=(2,2)\vec{a} - \vec{b} = (3, 4) - (1, 2) = (3-1, 4-2) = (2, 2)

While vector subtraction b⃗−a⃗\vec{b} - \vec{a} is:

b⃗−a⃗=(1,2)−(3,4)=(1−3,2−4)=(−2,−2)\vec{b} - \vec{a} = (1, 2) - (3, 4) = (1-3, 2-4) = (-2, -2)

Note that a⃗−b⃗≠b⃗−a⃗\vec{a} - \vec{b} \neq \vec{b} - \vec{a} in general. In fact, a⃗−b⃗=−(b⃗−a⃗)\vec{a} - \vec{b} = -(\vec{b} - \vec{a}).

Example of Vector Subtraction
Visualization of vector subtraction a - b = (3,4,0) - (1,2,0) = (2,2,0).

Applications of Vector Subtraction

Vector subtraction has many applications in real life:

  1. Calculating Displacement: If a⃗\vec{a} is the final position and b⃗\vec{b} is the initial position, then a⃗−b⃗\vec{a} - \vec{b} is the displacement vector.

  2. Calculating Distance: In games like Angry Birds, vector subtraction is used to calculate the distance and direction between the bird and the target.

  3. Physics: Vector subtraction is used to calculate the resultant force in systems with multiple forces.

Vector Subtraction to Find the Resultant

Vector subtraction can also be used to find the resultant vector. A resultant vector is a vector that represents the combined effect of two or more vectors.

Suppose an object moves from initial position p⃗1\vec{p}_1 to final position p⃗2\vec{p}_2. The displacement vector of the object is:

d⃗=p⃗2−p⃗1\vec{d} = \vec{p}_2 - \vec{p}_1

This resultant vector shows the direction and distance of the object's displacement.

Properties of Vector Subtraction

Vector subtraction has several important properties:

  1. a⃗−b⃗=a⃗+(−b⃗)\vec{a} - \vec{b} = \vec{a} + (-\vec{b})
  2. a⃗−a⃗=0⃗\vec{a} - \vec{a} = \vec{0} (zero vector)
  3. a⃗−0⃗=a⃗\vec{a} - \vec{0} = \vec{a}
  4. 0⃗−a⃗=−a⃗\vec{0} - \vec{a} = -\vec{a}
  5. a⃗−b⃗=−(b⃗−a⃗)\vec{a} - \vec{b} = -(\vec{b} - \vec{a})

Example Problem

A monkey is at position (3, 4) and wants to get a banana located at position (2, 1). Determine the displacement vector of the monkey to reach the banana.

Solution:

Monkey's position: m⃗=(3,4)\vec{m} = (3, 4) Banana's position: p⃗=(2,1)\vec{p} = (2, 1)

Displacement vector of the monkey to the banana:

d⃗=p⃗−m⃗=(2,1)−(3,4)=(2−3,1−4)=(−1,−3)\vec{d} = \vec{p} - \vec{m} = (2, 1) - (3, 4) = (2-3, 1-4) = (-1, -3)

Therefore, the monkey needs to move 1 unit in the negative x-axis direction and 3 units in the negative y-axis direction to reach the banana.

The magnitude of the displacement vector can be calculated using the Pythagorean theorem:

∣d⃗∣=(−1)2+(−3)2=1+9=10≈3.16|\vec{d}| = \sqrt{(-1)^2 + (-3)^2} = \sqrt{1 + 9} = \sqrt{10} \approx 3.16
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