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Here’s How Vector Subtraction Works

A vector is a term used to define any line segments with a specified starting and ending point. All vectors are drawn at an angle and have a specified direction. Learning to subtract vectors is helpful when you need to see how much one vector must travel to get to the other.

Vector subtraction is the process of subtracting the coordinates of one vector from the coordinates of a second vector.

See the example below. The coordinates of vector a are marked as (3,3) and the coordinates of vector b as (1, 2).

Graph showing the process of vector subtraction

When subtracting vectors, you must take the first vector quantities and subtract the second quantity. Let's subtract vector b from vector a:

Formula for vector subtraction process

Formula for vector subtraction process

Formula for vector subtraction process

Your resultant vector coordinates for this particular example are (2, 1).

Vector Addition, Magnitude, and Direction

Every vector is shaped like an arrow and has a magnitude and a direction. The actual length of the arrow, the amount traveled from one coordinate to the other, is the magnitude. The direction of the vector is the angle at which it’s pointing.

In the addition of vectors, you connect the tail of the vector (the ending coordinates) to the head of the vector (the beginning coordinates).

See the example below. The starting points are (4, 4) and (2, 3) and the terminal point is (6, 7)

Vector subtraction: Graph showing the process of vector addition

From this example, we can see that the magnitude of the resultant vector and the direction of the resultant vector differ from that of the initial points.

Breaking Down Vector Subtraction

When subtracting vectors, the direction of the vector being subtracted needs to be reversed. This indicates that the length of one vector is being subtracted from the other vector. See the example below. Vector v is shown going in the opposite direction to indicate that it’s being subtracted from the length of vector u. When you reverse the direction of a vector, you turn it into a negative vector.

Graph showing how to break down vector subtraction

The figure below shows coordinates (-6,1) being subtracted from (4, -2). Let’s do this vector subtraction process:

Graph showing coordinates (-6,1) being subtracted from (4, -2)

Formula showing coordinates (-6,1) being subtracted from (4, -2)

Formula showing coordinates (-6,1) being subtracted from (4, -2)

Formula showing coordinates (-6,1) being subtracted from (4, -2)

Formula showing coordinates (-6,1) being subtracted from (4, -2)

You can see that because we are subtracting the x-value of -6 from 4, the two negatives cancel out. It's as if you took the absolute value of the number, meaning that you removed all its negative values and turned it into a positive.

Mastering vector subtraction makes it easier to understand other trigonometry concepts. It gives you a better understanding of the difference between the magnitude and direction of a vector and how two negative values cancel out each other and result in a positive value.

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