# Unit Vectors

A vector has both magnitude and direction. A unit vector is a vector with magnitude of 1. In some situations it is helpful to find a unit vector that has the same direction as a given vector.

A unit vector of v, in the same direction as v, can be found by dividing v by its magnitude $\parallel v\parallel$.

UNIT VECTOR:

If $v\ne 0$ and $\parallel v\parallel$ represents the magnitude of vector v, then its unit vector u is:

$u=\frac{v}{\parallel v\parallel }=\frac{1}{\parallel v\parallel }v$

The unit vector u has a length of 1 in the same direction as v.

The unit vectors $〈1,0〉$ and $〈0,1〉$ are special unit vectors called standard unit vectors and are represented by the vectors i and j as follows:

$i=〈1,0〉$      $j=〈0,1〉$

Any vector in a plane can be written using these standard unit vectors.

This vector sum is called a linear combination. For example, vector $v=〈3,11〉=3i+11j$.

Let's look at some examples.

To work these examples requires the use of various differentiation rules. If you are not familiar with a rule go to the associated topic for a review.

Example 1: Find a unit vector u in the same direction as $v=〈12,-9〉$ and show that it has a magnitude of 1.
 Step 1: Find the magnitude of v. The magnitude of a vector is calculated by taking the square root of the sum of the squares of its components. $v=〈x,y〉$ $\parallel v\parallel =\sqrt{{x}^{2}+{y}^{2}}$ $v=〈12,-9〉$ ||v|| = $\sqrt{{12}^{2}+{\left(-9\right)}^{2}}$ $\parallel v\parallel =\sqrt{225}={15}$ Step 2: Calculate the unit vector. $u=\frac{v}{\parallel v\parallel }=\frac{1}{\parallel v\parallel }v$ $u=\frac{v}{\parallel v\parallel }=\frac{1}{\parallel v\parallel }v$ $u=\frac{〈12,-9〉}{15}=\frac{1}{15}〈12,-9〉$ $u=〈\frac{12}{15},-\frac{9}{15}〉={〈}\frac{4}{5}{,}{-}\frac{3}{5}{〉}$ Step 3: Show that the vector u has a magnitude of 1. The magnitude of a vector is calculated by taking the square root of the sum of the squares of its components. $v=〈x,y〉$ $\parallel v\parallel =\sqrt{{x}^{2}+{y}^{2}}$ $u=〈\frac{4}{5},-\frac{3}{5}〉$ ||u|| = ||u|| = $\parallel u\parallel =\sqrt{1}=1$
Example 2: If u = -3i + 2j and v = -i + 6j, find 2u + 4v.
 Step 1: Find vectors 2u and 4v using scalar multiplication. $2u=-6i+4j$ $4v=-4i+24j$ Step 2: Add vectors 2u and 4v using vector addition. 2u + 4v (-6i + 4j) + (-4i + 24j) (-6i - 4i) + (4j + 24j) $\left(-10i+28j\right)$

 Related Links: Math algebra Direction Angles of Vectors The Dot Product Pre Calculus

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