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

∥ v ∥

UNIT VECTOR:

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

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

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.

v = 〈 v_{1}, v_{2} 〉 = v_{1} i + v_{2} j \to L i n e a r C o m b i n a t i o n

This vector sum is called a linear combination. For example, vector

v = 〈 3, 11 〉 = 3 i + 11 j

.

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.

$v = 〈 x, y 〉$

$∥ v ∥ = \sqrt{x^{2} + y^{2}}$

$v = 〈 12, - 9 〉$

||v|| = $\sqrt{12^{2} + {(- 9)}^{2}}$

$∥ v ∥ = \sqrt{225} = 15$

Step 2: Calculate the unit vector.

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

$u = \frac{〈 12, - 9 〉}{15} = \frac{1}{15} 〈 12, - 9 〉$

$u = 〈 \frac{12}{15}, - \frac{9}{15} 〉 = 〈 \frac{4}{5}, - \frac{3}{5} 〉$

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.

$v = 〈 x, y 〉$

$∥ v ∥ = \sqrt{x^{2} + y^{2}}$

$u = 〈 \frac{4}{5}, - \frac{3}{5} 〉$

||u|| = $\sqrt{{(\frac{4}{5})}^{2} + {(- \frac{3}{5})}^{2}}$

||u|| = $\sqrt{\frac{16}{25} + \frac{9}{25}} = \sqrt{\frac{25}{25}}$

$∥ u ∥ = \sqrt{1} = 1$

Example 2: If u = -3i + 2j and v = -i + 6j, find 2u + 4v.

Step 1: Find vectors 2u and 4v using scalar multiplication.

$k v = k v_{1}, v_{2} = k v_{1}, k v_{2} \to S c a l a r M u l t i p l i c a t i o n$

$2 u = 2 (- 3 i + 2 j) = [2 \cdot (- 3 i) + 2 \cdot 2 j]$

$2 u = - 6 i + 4 j$

$4 v = 4 (- i + 6 j) = [4 \cdot (- i) + 4 \cdot 6 j]$

$4 v = - 4 i + 24 j$

Step 2: Add vectors 2u and 4v using vector addition.

$u + v = u_{1} + v_{1}, u_{2} + v_{2} \to V e c t o r A d d i t i o n$

2u + 4v

(-6i + 4j) + (-4i + 24j)

(-6i - 4i) + (4j + 24j)

$(- 10 i + 28 j)$

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

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