## Using Components To Add And Subtract Vectors

Another way of adding vectors is to add the components. Previously, we saw that vectors can be expressed in terms of their horizontal and vertical components. To add vectors, merely express both of them in terms of their horizontal and vertical components and then add the components together.

**Vector with Horizontal and Vertical Components**: The vector in this image has a magnitude of 10.3 units and a direction of 29.1 degrees above the x-axis. It can be decomposed into a horizontal part and a vertical part as shown.

For example, a vector with a length of 5 at a 36.9 degree angle to the horizontal axis will have a horizontal component of 4 units and a vertical component of 3 units. If we were to add this to another vector of the same magnitude and direction, we would get a vector twice as long at the same angle. This can be seen by adding the horizontal components of the two vectors and the two vertical components . These additions give a new vector with a horizontal component of 8 and a vertical component of 6 . To find the resultant vector, simply place the tail of the vertical component at the head of the horizontal component and then draw a line from the origin to the head of the vertical component. This new line is the resultant vector. It should be twice as long as the original, since both of its components are twice as large as they were previously.

## Adding And Subtracting Vectors

One of the ways in which representing physical quantities as vectors makes analysis easier is the ease with which vectors may be added to one another. Since vectors are graphical visualizations, addition and subtraction of vectors can be done graphically.

The graphical method of vector addition is also known as the head-to-tail method. To start, draw a set of coordinate axes. Next, draw out the first vector with its tail at the origin of the coordinate axes. For vector addition it does not matter which vector you draw first since addition is commutative, but for subtraction ensure that the vector you draw first is the one you are subtracting *from*. The next step is to take the next vector and draw it such that its tail starts at the previous vectorâs head . Continue to place each vector at the head of the preceding one until all the vectors you wish to add are joined together. Finally, draw a straight line from the origin to the head of the final vector in the chain. This new line is the vector result of adding those vectors together.

**Graphical Addition of Vectors**: The head-to-tail method of vector addition requires that you lay out the first vector along a set of coordinate axes. Next, place the tail of the next vector on the head of the first one. Draw a new vector from the origin to the head of the last vector. This new vector is the sum of the original two.

## How Do You Find The Unit Vector Perpendicular To Two Vectors

The cross-product of two non-parallel results in a vector that is a vector that is perpendicular to both of them. So, for the given two vectors \ and \, we know that, \ will be a vector that is perpendicular to both \. Further, we find the unit vector of this resultant vector to obtain the unit vector perpendicular to the two given vectors.

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## Application Of Unit Vector

Unit vectors specify the direction of a vector. Unit vectors can exist in both two and three-dimensional planes. Every vector can be represented with its unit vector in the form of its components. The unit vectors of a vector are directed along the axes. Unit vectors in 3-d space can be represented as follows: **v** = x^ + y^ + z^.

In the 3-d plane, the vector **v **will be identified by three perpendicular axes . In mathematical notations, the unit vector along the x-axis is represented by i^. The unit vector along the y-axis is represented by j^, and the unit vector along the z-axis is represented by k^.

The vector v can hence be written as:

**v** = xi^ + yj^ + zk^

Electromagnetics deals with electric forces and magnetic forces. Here vectors are come in handy to represent and perform calculations involving these forces. In day-to-day life, vectors can represent the velocity of an airplane or a train, where both the speed and the direction of movement are needed.

## Properties Of A Unit Vector

The contents of this topic can be summarized in the properties of a unit vector in the following manner:

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## Define Unit Vector Class 11

A unit vector definition in Physics is, a unit vector is a direction vector. A unit vector symbol is similar to the mathematical symbol of the exponent, and that is ^. Unit vector symbol in Physics is pronounced as cap or hat.

For calculating the magnitude of any given vector, we use the coordinate system as follows:

Here,

A unit vector, i cap indicates the direction of an object along the x-axis.

A unit vector, j cap indicates the direction of an object along the y-axis.

A unit vector, k cap indicates the direction of an object along the z-axis.

So, do you know what is unit vector in Physics?

Well! We define unit vector in Physics with the following equation:

Unit vector = \

So, a unit vector \ having the same direction as a vector \ is given as

\ = \ |\|

Here,

\ = a unit vector

\ = represents the vector

|\| = represents the magnitude of the vector

## Unit Vector Vs Basis Vector

When reading about vectors I sometimes have seen unit vectors multiplied by the components and other times I’ve seen basis vectors used instead. $$v=x \hat i+y \hat j+z \hat k$$

$$v=xe_x+ye_y+ze_z$$Occasionally, I’ve seen both used in a single source. As far as I can tell, they seen to be doing the same thing, i.e., showing what direction each component is pointing while not changing the numerical value of any of the components .

My questions are:What is the difference between an unit vector and a basis vector? And are they interchangeable in specifying the directions of components?

- 1$\begingroup$It’s worth pointing out that it doesn’t really make sense to talk about ‘a basis vector’:
*any*nonzero vector is part of some basis , so it’s not really helpful. Rather you need to talk about a particular basis, which is a set of vectors with nice properties of which the vector of interest is one, and this is what people mean usually. It*is*useful to talk about a unit vector since that is defined as a property of the vector on its own.$\endgroup$ user107153

A unit vector $v$ is a vector whose norm is unity: $||v||=1$. That’s all. Any non-zero vector $w$ can define a unit vector $w/||w||$.

For example, $$ and $$ form a basis of the plane . So both $$ and $$ are basis vectors. $$ is a unit vector, but not a basis vector in that case. But you could also consider another basis made of $$ and $$, then $$ would also be a unit vector.

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## Radial And Normal Unit Vectors

In addition to **i**, **j**, and **k**, two other unit vectors are used in this text. A vector of unit magnitude in the radially outward direction is designated by r is formed by taking the position vector **r** and dividing it by its magnitude:

unit magnitude but preserves the radial direction of **r.** Unlike the three Cartesian unit vectors, r may vary in direction because the position vector **r** varies in direction.

The unit vector n ^ is normal, or perpendicular, to a surface at a given point . For the special case of a spherical surface the normal vector n

Figure 2.25. The unit vector r points radially outward. The unit vector n is normal to the surface.

Questions

- 6.

How can the direction cosines be negative? Describe the orientation of a vector having all three direction cosines negative.

Sarhan M. Musa, in, 2016

## What Is Unit Vector In Physics

A vector quantity bears both magnitude and direction. For example, displacement is the shortest distance taken to reach your destination. It is displayed as a displacement vector.

A vector also carries a negative magnitude when an object takes the opposite direction. There is one more term called a unit vector. Unit vectors have a magnitude of 1.

A speciality of a unit vector in Physics is, a vector can be represented in the space using a unit vector.

On this page, we will understand what is unit vector in Physics Class 11, and the unit vector definition in Physics in detail.

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## Vectors Vs Unit Vectors

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grzz said:A UNIT vector has magnitude of 1.

- 898

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Ascendant78 said:Actually, I think I might get it now…If I am understanding right, say if I calculated a unit vector force of a gravitational field between two objects to be say 500N and used meters for the units, then if they were 10m away from each other when I calculated this force, it means the force between them is 500N per m, making the overall force 5000N, or is that wrong?

Ascendant78 said:Oh, I get their relation and the concept behind them in that a unit vector is 1 of whatever quantity is being measured. What I don’t understand is exactly what they tell you relative to one another? Say if I found the vector force of gravity from a planet, then found the unit vector force of gravity from that same planet, I’m not sure of what each one would be telling me relative to each other? Like I said before, I only ever used the vector force of gravity, I never used the unit vector force, so I don’t know what that would tell me about the gravitational force?

*velocity**velocity*

## Some Special Cases Of Vector Addition

The following are some special cases to make vector calculation easier to represent.

**1. =0° **: Here is the angle between the two vectors. That is, if the value of is zero, the two vectors are on the same side. In this case, the value of the resultant vector will be

Thus, the absolute value of the resultant vector will be equal to the sum of the absolute values of the two main vectors. Notice the image below

**2. =180° **: Here, if the angle between the two vectors is 180°, then the two vectors are opposite to each other. That is, the value of cos here will be -1.

Here the absolute value of the resultant vector is equal to the absolute value of the subtraction of the two vectors. And the resultant vector will be oriented towards it whose absolute value is higher than the others.

**3. a=b and =180° **: Here the two vectors are of equal value and are in opposite directions to each other. In this case, the absolute value of the resultant vector will be zero. That is, the resolution vector is a null vector

**2. =90° **: If the angle between the two vectors is 90 degrees. The value of **cos** will be zero. So, here the resultant vector will follow the formula of Pythagoras

In this case, the two vectors are perpendicular to each other. And the resultant vector will be located at the specified angle with the two vectors. Suppose, here two vectors **a**, **b** are taken and the resultant vector **c** is located at angle with **a** vector Then the direction of the resultant vector will be

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## What Is The Origin Of Unit Vector Notation

- 2$\begingroup$I’m retagging your question, since it is really about history $\endgroup$Oct 24 ’11 at 4:16
- 5 Theo BuehlerOct 24 ’11 at 4:43
- $\begingroup$While this is a mildly interesting historical question, I do not think MO should become a repository for asking ‘where does this common notation come from?’ style questions. Searching on google for “history of mathematical notation” gives a number of interesting pages.$\endgroup$Oct 24 ’11 at 6:41
- 1

Maybe it originates from Hamilton’s quaternions $\mathbb$, which has a basis $1,i,j,k$ as a real vector space, and the multiplications there, namely, $i\cdot j=k, j\cdot k=i, k\cdot i=j$ correspond exactly to the wedge product in $\mathbb^3$. So $\mathbb^3$ can be viewed as the imanginary part of $\mathbb$.

Anyway, this is just my understanding or my guess.

## What Are Vectors In Physics

A physical quantity is a quantity whose physical properties you can measure. Such as mass, force, velocity, displacement, temperature, etc.

Suppose you are told to measure your happiness. In this case, you can never measure your happiness. That is, you cannot describe and analyze with measure how much happiness you have. So, happiness here is not a physical quantity.

Suppose you have a fever. And the doctor ordered you to measure your body temperature. Then you measured your body temperature with a thermometer and told the doctor. When you tell your doctor about your body temperature, you need to use the word degree centigrade or degree Fahrenheit.

So, the temperature here is a measurable quantity. So we will use temperature as a physical quantity.

In general, we will divide the physical quantity into three types

In this tutorial, we will only discuss vector quantity. But before that, lets talk about scalar.

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## Multiplying Vectors By A Scalar

*vector*by a scalar changes the magnitude of the

*vector*but not the direction.

*unit*-less scalar and multiply it to a

*vector*, you get a version of the original

*vector*which is 3 times as long.

*units*used in

*vector*quantities are intrinsically scalars multiplied by the

*vector*.

*unit*of meters per second used in velocity, which is a

*vector*, is made up of two scalars, which are magnitudes: the scalar of length in meters and the scalar of time in seconds.

*unit*

*vector*in a particular direction by these scalars.

## Finding Unit Vectors In Three

In the three-dimensional plane, the vector **v **will be represented by three perpendicular axes, namely the x, y, and the z-axis. In mathematical notations, the unit vector along the x-axis is represented by i^. The unit vector along the y-axis is represented by j^, and the unit vector along the z-axis is represented by k^.

The vector v can hence be written as:

**v** = xi^ + yj^ + zk^

For writing these unit vectors in planar forms, we can make use of parentheses such as below:

Hence, we can rewrite the formula for finding the unit vector in two ways:

**u** = **v** / |v|

**u** = **v** / ^2 + ^2 +^2)

Or we can also write it as:

**u **= / ^2 + ^2 + ^2)

*Example 4*

For the vector v = , find its unit vector.

Solution

We are familiar with the formula for finding the unit vector:

**u** = **v** / |v|

So, by inserting the necessary parameters:

**u** = / ^2 + ^2 + ^2)

**u** = / 169

**u** = / 13

So **u** is the required unit vector.

Proof

For proof, lets find the magnitude of **u:**

|u| = ^2 + ^2 + ^2)

|u| =

Hence, it is proved that the vector** u** is the unit vector of **v.**

*Example 5*

Find the unit vector of the vector v = .

Solution

The formula for finding the unit vector is stated as:

**u** = **v** / |v|

So, by inserting the necessary parameters:

**u** = / ^2 + ^2 + ^2)

**u** = / 21

So **u** is the required unit vector.

Proof

For proof, lets find the magnitude of **u:**

|u| = ^2 + ^2 + ^2)

|u| = 0.99

Hence, it is proved that the vector** u** is the unit vector of **v.**

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## Matter Exists In Space And Time

*vectors*,

*vectors*spaces and

*vector*algebra is needed to understand his ideas.

*unit*

*vector*basis.

*Vector*analysis is methodological.

*vector*has a component and a magnitude-direction form.

*vectors*and calculus because he needed that mathematics.

*Vectors*can be used to represent physical quantities.

*Vectors*are a combination of magnitude and direction, and are drawn as arrows.

*vectors*are constructed this way, it is helpful to analyze physical quantities as

*vectors*.

*vector*that represents a magnitude of 100, one may draw a line that is 5

*units*long at a scale of $\displaystyle \frac$.

*vector*, the magnitude is only important as a way to compare two

*vectors*of the same

*units*.

*vectors*that are perpendicular to any plane.

*units*of angular velocity are radians per second.

*unit*time.

*vector*diagram illustrating circular motion.

*vector*is the angular velocity

*vector*, pointing perpendicular to the plane of motion and with magnitude equal to the instantaneous velocity.

*vectors*have a length, called the magnitude, which represents some quality of interest so that the

*vector*may be compared to another

*vector*.