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What Does Unit Vector Mean?

A unit vector for the vector (4, 3)

Unit vectors are vectors with a length of 1. All vectors have a corresponding unit vector. You can find it by using this formula:

Formula

Unit Vector

(x1,y1) unit vector = 1 | (x1,y1)| (x1,y1)

You also have unit vectors that sit along the axes. All vectors in the plane are made up of a combination of these.

Theory

The Unit Vectors Along the Coordinate Axes

  • The unit vector along the x-axis: ex = (1, 0).

  • The unit vector along the y-axis: ey = (0, 1).

A vector can always be expressed using the unit vectors ex and ey like this:

(a,b) = aex + bey

Example 1

Find the unit vector of (3, 4).

(3, 4) unit vector = 1 | (3, 4)| (3, 4) = 1 32 + 42 (3, 4) = 1 25 (3, 4) = 1 5 (3, 4) = (3 5, 4 5)

Example 2

Write the vector (3, 5) by using the unit vectors that sit along the two axes.

You know that the unit vectors of the axes are ex and ey, and that those are ex = (1, 0) and ey = (0, 1). That means you can decompose the vector and find the expression you’re looking for. It will look like this:

(3, 5) = (3, 0) + (0, 5) = 3 (1, 0) + 5 (0, 1) = 3 ex + 5 ey

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