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Wize University Linear Algebra Textbook > Vectors

Unit Vectors

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Unit Vectors

A unit vector is a vector that has length one.

Using the notion of norm, we say v⃗\vec v is a unit vector if and only if ∥v⃗∥=1\|\vec{v}\|= 1.

How to Find a Unit Vector

The unit vector in the same direction as a given vector v⃗\vec{v} is given by:
1∥v⃗∥v⃗\boxed{\quad \dfrac{1}{\lVert \vec{v}\rVert}\vec{v} \quad}
Steps
  1. Find the length ∥v⃗∥\lVert \vec{v} \rVert
  2. Multiply v⃗\vec v by the scalar 1∥v⃗∥\dfrac{1}{\lVert\vec{v}\rVert}(divide by its length) to obtain 1∥v⃗∥v⃗\dfrac{1}{\lVert \vec{v}\rVert}\vec{v}
*This is sometimes called normalizing a vector.
Example 1
Find the unit vector in the same direction as the vector v⃗=⟨−3,−4⟩\vec v = \lang -3, -4 \rang.

∥v⃗∥=(−3)2+(−4)2=9+16=25=5\lVert \vec v \rVert = \sqrt{(-3)^2 + (-4)^2} = \sqrt{9+16} = \sqrt{25} = 5
The unit vector in the same direction as v⃗\vec v is therefore:
1∥v⃗∥v⃗  =  15⟨−3,−4⟩  =  ⟨−35,−45⟩\dfrac{1}{\lVert \vec v \rVert} \vec v \ \ = \ \ \dfrac{1}{5} \lang -3, -4 \rang \ \ = \ \ \boxed{\left\lang -\dfrac{3}{5}, -\dfrac{4}{5} \right\rang}
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Standard Basis Vectors

Standard basis vectors are unit vectors that run along each axis (positive side only).

In R2\reals^2:
  • i^=⟨1, 0⟩\hat{i}=\lang 1,\ 0\rang runs along the positive xx-axis
  • j^=⟨0, 1⟩\hat{j}=\lang 0,\ 1\rang runs along the positive yy-axis


In R3\reals^3:
  • i^=⟨1, 0, 0⟩\hat{i}=\lang 1,\ 0,\ 0\rang runs along the positive xx-axis
  • j^=⟨0, 1, 0⟩\hat{j}=\lang 0,\ 1,\ 0\rang runs along the positive yy-axis
  • k^=⟨0, 0, 1⟩\hat{k}=\lang 0,\ 0,\ 1\rang runs along the positive zz-axis
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Decomposing Vectors with Unit Vectors

When plotting a point on a graph, we usually count along each axis one at a time. We are adding multiples of unit vectors!

Example 2

v⃗=⟨3,−1,2⟩\vec{v}=\lang 3,−1,2 \rang ==
3
i^\hat i +
-1
j^\hat j +
2
k^\hat k
This is another notation for writing vectors!
Try writing out this expression with column vectors:
v⃗=[3−12]=  3[100]  +  (−1)[010]  +  2[001]\vec{v} = \begin{bmatrix} 3\\ -1\\ 2 \end{bmatrix} = \ \ 3 \begin{bmatrix} 1\\ 0\\ 0 \end{bmatrix} \ \ + \ \ (-1) \begin{bmatrix} 0\\ 1\\ 0 \end{bmatrix} \ \ + \ \ 2 \begin{bmatrix} 0\\ 0\\ 1 \end{bmatrix}
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Example: Unit Vectors

Part A)

Find the unit vector in the same direction as v⃗=⟨1,2,−2⟩\vec v = \lang 1,2,-2 \rang.
First calculate ∥v⃗∥\lVert\vec{v}\rVert:
∥v⃗∥=12+(2)2+(−2)2=9=3\lVert\vec{v}\rVert = \sqrt{1^2 + (2)^2 + (-2)^2} = \sqrt{9} = 3
Then the unit vector in the same direction as v⃗\vec v is given by:
1∥v⃗∥v⃗=13⟨1,2,−2⟩=⟨13, 23, −23⟩\dfrac{1}{\lVert\vec{v}\rVert}\vec{v} = \dfrac{1}{3} \lang 1, 2, -2\rang =\boxed{ \left\lang\dfrac{1}{3}, \ \dfrac{2}{3}, \ -\dfrac{2}{3}\right\rang }

Part B)

Write the answer to Part A) as a linear combination of the standard basis vectors i^, j^, k^∈R3\hat i, \ \hat j, \ \hat k \in \reals^3.
Recall that the standard basis vectors in R3\reals^3 are defined to be:
i^=[100],  j^=[010],  k^=[001]\hat i = \begin{bmatrix} 1\\ 0\\ 0 \end{bmatrix} , \ \ \hat j = \begin{bmatrix} 0\\ 1\\ 0 \end{bmatrix} , \ \ \hat k = \begin{bmatrix} 0\\ 0\\ 1 \end{bmatrix}
So we can rewrite the vector ⟨13, 23,−23⟩\left\lang\dfrac{1}{3}, \ \dfrac{2}{3}, -\dfrac{2}{3}\right\rang as follows:
[   1/3   2/3−1/3]=  13[100]  +  23[010]  −  23[001]=  13 i^ + 23 j^ − 23 k^\begin{aligned} \begin{bmatrix} {\ \ \ 1}/{3}\\ {\ \ \ 2}/{3}\\ -{1}/{3} \end{bmatrix} &= \ \ \dfrac{1}{3} \begin{bmatrix} 1\\ 0\\ 0 \end{bmatrix} \ \ + \ \ \dfrac{2}{3} \begin{bmatrix} 0\\ 1\\ 0 \end{bmatrix} \ \ - \ \ \dfrac{2}{3} \begin{bmatrix} 0\\ 0\\ 1 \end{bmatrix}\\[2em] &= \ \ \dfrac{1}{3} \ \hat i \ + \ \dfrac{2}{3} \ \hat j \ - \ \dfrac{2}{3} \ \hat k \end{aligned}

Practice: Unit Vectors

Find the value of kk such that u⃗=⟨k,12,−k⟩\vec u = \left\lang k, \dfrac{1}{\sqrt2}, -k \right\rang is a unit vector.
Extra Practice
Find the unit vector in the opposite direction as 𝑣⃗ =(1,−1,0,−1,0)𝑣⃗\ =(1,−1,0,−1,0)
Find the unit vector in the same direction as v⃗=(1, 0, −3, 4, −2)\vec{v}=\left(1,\ 0,\ -3,\ 4,\ -2\right).
Vectors in 2D and 3D Systems (2)
Practice
Find the unit vector that points in the same direction as a⃗ =<−10, 4, −3>\vec{a}~=<-10,~4,~-3>.
Unit Vectors
Practice: Unit Vectors
Find the value of kk such that u⃗=⟨k,12,−k⟩\vec u = \left\lang k, \dfrac{1}{\sqrt2}, -k \right\rang is a unit vector.
133 - FML 3 - 18.1W e.g. 26
Find on expression for a vector of unit length in the direction of v⃗=<v1,v2,v3>\bcb{\vec{v} = \left< v_1, v_2, v_3 \right>}.
Practice: Unit Vector
Practice: Unit Vectors
Find the unit vector in the same direction as v⃗=(1,−2,−1)\vec v = (1,-2,-1).
Practice: Unit Vector
Practice: Unit Vector
Find the unit vector in the same direction as i⃗−5k⃗\vec i-5\vec k.
Practice: Unit Vector
Practice: Unit Vector
Find the unit vector in the same direction as i^−5k^\hat{i}-5\hat{k}.
Concept Clarifier
Find the unit vector in the direction of v⃗=(3,4,5)\vec{v}=(3,4,5)
Practice: Cross Product
Practice: Cross Product
Consider u⃗=[−2, 3, −1]\vec{u}=\left[-2,\ 3,\ -1\right] and v⃗=[1, 0, −1]\vec{v}=\left[1,\ 0,\ -1\right].
133 - FML 3 - 18.1W e.g. 24
If a vector v⃗\bcb{\vec{v}} has length L\bcb{\mathcal{L}}, give an expression for a unit vector in the same direction as v⃗\bcb{\vec{v}}.
133 - FML 3 - 18.1W e.g. 26
Find an expression for a vector of unit length in the direction of v⃗=<v1,v2,v3>\bcb{\vec{v} = \left< v_1, v_2, v_3 \right>}.
133 - FML 3 - 18.1W e.g. 48
Find a unit vector in the direction of the projection of u⃗=<102,112,−56>\bcb{\vec{u} = \left< 102, 112, -56 \right>} onto v⃗=<0,3,4>\bcb{\vec{v} = \left< 0, 3, 4 \right>}.
133 - FML 3 - 18.1W e.g. 24
If a vector v⃗\bcb{\vec{v}} has length L\bcb{\mathcal{L}}, give an expression for a unit vector in the same direction as v⃗\bcb{\vec{v}}.
$\tkct{cut from 19.4F}$ Mid $\tkco{ S}$ | 133 - FML 3 - 18.1W e.g. 48
Find a unit vector in the direction of the projection of u⃗=<102,112,−56>\bcb{\vec{u} = \left< 102, 112, -56 \right>} onto v⃗=<0,3,4>\bcb{\vec{v} = \left< 0, 3, 4 \right>}.
Concept Clarifier
Find the unit vector in the direction of v⃗=(3,4,5)\vec{v}=(3,4,5)
Unit Vector Practice Question
Find the unit vector in the same direction as v⃗=(3,4,5)\vec{v}=(3,4,5)
Practice Question: Unit Vector
Practice Question: Unit Vector
Find the unit vector in the same direction as 𝑣 = (1, −2, −1).
Practice Question: Unit Vector
Practice Question: Unit Vector
Find the unit vector in the same direction as 𝑣 = (1, −2, −1).
Practice Question: Unit Vector
Practice Question: Unit Vector
Which one of the following is not a unit vector?
Practice Question: Unit Vector
Practice Question: Unit Vector
Find the unit vector in the opposite direction as 𝑣⃗ =(1,−1,0,−1,0)𝑣⃗\ =(1,−1,0,−1,0).
Practice: Unit Vector
Practice: Unit Vector
Find the unit vector in the same direction as 𝑣 = (1, −2, −1).
Practice: Unit Vector
Practice: Unit Vector
Find the unit vector in the same direction as i⃗−5k⃗\vec i-5\vec k.
$\tkcth{Moved }\bcth{\to} \key{ Mid} \tkco{ S } \tkcth{ch 3 quiz ✓}$ |
A unit vector parallel to the vector v⃗=<4, −3>\vv = \rowvec{4}{-3} is:
$\tkcth{Moved }\bcth{\to} \key{ Mid} \tkco{ S } \tkcth{ch 3 quiz ✓}$ |
A unit vector perpendicular to the vector w⃗=<2, 5>\vw = \rowvec{2}{5} is:
$\tkcth{Moved }\bcth{\to} \key{ Mid} \tkco{ S } \tkcth{ch 3 quiz ✓}$ |
Find a vector parallel to the vector u⃗=<1, 4, 8>\vu = \rowvecth{1}{4}{8} whose magnitude is 1.
DIY Unit Vector
Find a vector whose length is one, that points in the same direction as the vector u⃗=<2, 1, −2>\vu = \rowvecth{2}{1}{-2}
Unit Vectors: Dot Product
Consider the vectors u=i−4k\bm{u}=\bm{i}-4\bm{k} and v=3i−2j+k\bm{v}=3\bm{i}-2\bm{j}+\bm{k} in R3\reals^3. Find (if possible) 3v⋅(k−2u)3\bm{v}\cdot(\bm{k}-2\bm{u}).