## 17Calculus - Unit Vectors

Unit vectors are defined as vectors whose length is exactly one. This means you can use a unit vector to define direction and then assign a length to the vector to get a unique vector.

The Standard Unit Vectors

There are three standard unit vectors listed in the table below and shown in the figure. source Wikipedia

The Three Standard Unit Vectors

Vector

Definition

$$\hat{i}$$

the unit vector in the direction of the x-axis

$$\hat{j}$$

the unit vector in the direction of the y-axis

$$\hat{k}$$

the unit vector in the direction of the z-axis

Every vector in 3-dim space can be described as a linear combination of these three standard unit vectors. You will almost always see the same letters ('i', 'j' and 'k') used to indicate the standard unit vectors. However, you may see them written either as bold letters or with an 'arrow' instead of 'hat', i.e. $$\hat{i} = \vec{i}$$.
Note - We recently ran across a book that used a little different notation. This book used $$\hat{x}$$, $$\hat{y}$$ and $$\hat{z}$$. Also, some books use the 'hat' notation to always indicate unit vectors.

You now have two ways to describe a vector, $$\vec{v} = \langle v_1, v_2, v_3 \rangle$$ or the same vector can be written $$\vec{v} = v_1 \hat{i} + v_2 \hat{j} + v_3 \hat{k}$$.

Okay, so let's watch a video explaining this in more detail. He goes through an example and explains this whole idea of the standard unit vectors as he works it.

### Khan Academy - standard unit vectors [9mins-53secs]

Finding a Unit Vector in a Specific Direction

If you have a vector, let's call it $$\vec{v}$$, and you want to find a unit vector in the same direction as $$\vec{v}$$, you just multiply the vector by the scalar $$1/ \norm{\vec{v}}$$. Said another way, you divide the vector by its length. This gives you the unit vector, $$\vec{u}$$, in the same direction as vector $$\vec{v}$$.
This is written as $$\displaystyle{ \vec{u} = \frac{\vec{v}}{\norm{\vec{v}}} }$$.

You will often see a $$\vec{u}$$ used to define a unit vector but not always. So make sure to check the context to determine exactly what is meant and keep track of unit vectors. They are important to vector calculus.

Okay, time for some practice problems. After that, your next topic is the dot product.

Practice

Find the unit vector in the direction of $$\langle 3,4 \rangle$$.

Problem Statement

Find the unit vector in the direction of $$\langle 3,4 \rangle$$.

Solution

### 1233 video

video by PatrickJMT

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Find the unit vector in the direction of $$\vec{w}=\langle 1/2,1/8 \rangle$$.

Problem Statement

Find the unit vector in the direction of $$\vec{w}=\langle 1/2,1/8 \rangle$$.

Solution

### 1234 video

video by PatrickJMT

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Find the unit vector in the direction of $$\vec{w}=\langle 0,5 \rangle$$.

Problem Statement

Find the unit vector in the direction of $$\vec{w}=\langle 0,5 \rangle$$.

Solution

### 1235 video

video by PatrickJMT

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Find the sum of $$3 \hat{i} + 5 \hat{j}$$ and $$2 \hat{i} - 7 \hat{j}$$.

Problem Statement

Find the sum of the vectors $$3 \hat{i} + 5 \hat{j}$$ and $$2 \hat{i} - 7 \hat{j}$$.

Solution

### 1218 video

video by Krista King Math

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Calculate $$\vec{a} + \vec{b}$$ for $$\vec{a} = -3\hat{i} + 2\hat{j}$$, $$\vec{b} = 2\hat{i} + 4\hat{j}$$.

Problem Statement

Calculate $$\vec{a} + \vec{b}$$ for $$\vec{a} = -3\hat{i} + 2\hat{j}$$, $$\vec{b} = 2\hat{i} + 4\hat{j}$$.

Solution

### 1236 video

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You CAN Ace Calculus

Wikipedia: Unit Vector

### Trig Formulas

The Unit Circle

The Unit Circle [wikipedia] Basic Trig Identities

Set 1 - basic identities

$$\displaystyle{ \tan(t) = \frac{\sin(t)}{\cos(t)} }$$

$$\displaystyle{ \cot(t) = \frac{\cos(t)}{\sin(t)} }$$

$$\displaystyle{ \sec(t) = \frac{1}{\cos(t)} }$$

$$\displaystyle{ \csc(t) = \frac{1}{\sin(t)} }$$

Set 2 - squared identities

$$\sin^2t + \cos^2t = 1$$

$$1 + \tan^2t = \sec^2t$$

$$1 + \cot^2t = \csc^2t$$

Set 3 - double-angle formulas

$$\sin(2t) = 2\sin(t)\cos(t)$$

$$\displaystyle{ \cos(2t) = \cos^2(t) - \sin^2(t) }$$

Set 4 - half-angle formulas

$$\displaystyle{ \sin^2(t) = \frac{1-\cos(2t)}{2} }$$

$$\displaystyle{ \cos^2(t) = \frac{1+\cos(2t)}{2} }$$

Trig Derivatives

 $$\displaystyle{ \frac{d[\sin(t)]}{dt} = \cos(t) }$$ $$\displaystyle{ \frac{d[\cos(t)]}{dt} = -\sin(t) }$$ $$\displaystyle{ \frac{d[\tan(t)]}{dt} = \sec^2(t) }$$ $$\displaystyle{ \frac{d[\cot(t)]}{dt} = -\csc^2(t) }$$ $$\displaystyle{ \frac{d[\sec(t)]}{dt} = \sec(t)\tan(t) }$$ $$\displaystyle{ \frac{d[\csc(t)]}{dt} = -\csc(t)\cot(t) }$$

Inverse Trig Derivatives

 $$\displaystyle{ \frac{d[\arcsin(t)]}{dt} = \frac{1}{\sqrt{1-t^2}} }$$ $$\displaystyle{ \frac{d[\arccos(t)]}{dt} = -\frac{1}{\sqrt{1-t^2}} }$$ $$\displaystyle{ \frac{d[\arctan(t)]}{dt} = \frac{1}{1+t^2} }$$ $$\displaystyle{ \frac{d[\arccot(t)]}{dt} = -\frac{1}{1+t^2} }$$ $$\displaystyle{ \frac{d[\arcsec(t)]}{dt} = \frac{1}{\abs{t}\sqrt{t^2 -1}} }$$ $$\displaystyle{ \frac{d[\arccsc(t)]}{dt} = -\frac{1}{\abs{t}\sqrt{t^2 -1}} }$$

Trig Integrals

 $$\int{\sin(x)~dx} = -\cos(x)+C$$ $$\int{\cos(x)~dx} = \sin(x)+C$$ $$\int{\tan(x)~dx} = -\ln\abs{\cos(x)}+C$$ $$\int{\cot(x)~dx} = \ln\abs{\sin(x)}+C$$ $$\int{\sec(x)~dx} =$$ $$\ln\abs{\sec(x)+\tan(x)}+C$$ $$\int{\csc(x)~dx} =$$ $$-\ln\abs{\csc(x)+\cot(x)}+C$$

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 The Standard Unit Vectors Finding a Unit Vector in a Specific Direction Practice

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