Basic Concept of Vector Subtraction
Vector subtraction gives the difference between two vectors. When we subtract from , we obtain the vector that must be added to to produce .
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Vector subtraction gives the difference between two vectors. When we subtract from , we obtain the vector that must be added to to produce .
Mathematically, vector subtraction is defined as:
In other words, subtracting from is equivalent to adding to the negative of .
The following four steps show two equivalent constructions of :
The visualization shows both constructions. The two arrows labeled have the same components, length, and direction even though they begin at different points.
Algebraically, subtract corresponding components. Suppose we have two vectors:
Then vector subtraction can be calculated as:
For two-dimensional vectors, the equation becomes:
Suppose there are two vectors:
Vector subtraction is:
While vector subtraction is:
Notice that in general. Reversing the order reverses the direction: .
Vector subtraction appears whenever we compare two vector quantities:
Calculating Displacement: If is the final position and is the initial position, then is the displacement vector.
Calculating Distance: In a game, subtracting the bird's position from the target's position gives the direction toward the target. The magnitude of that difference vector gives the straight-line distance.
Comparing Velocities: In physics, the velocity of object relative to object is . Reversing the order reverses the relative-velocity vector.
The displacement between two positions is found by subtracting the initial position vector from the final position vector.
Suppose an object moves from initial position to final position . The displacement vector of the object is:
This vector points from the initial position to the final position. Its magnitude is the straight-line distance between those positions, which may be shorter than the path actually traveled.
Vector subtraction satisfies the following identities:
A monkey is at position and wants to get a banana located at position . Determine the displacement vector of the monkey to reach the banana.
Solution:
Monkey's position:
Banana's position:
Displacement vector of the monkey to the banana:
The monkey needs to move in the negative -axis direction and in the negative -axis direction to reach the banana.
The magnitude of the displacement vector can be calculated using the Pythagorean theorem:
Published: . Updated: .