Vectors <!--Induction_and_Inductance -->

General Physics (calculus based) Class Notes

Dr. Rakesh Kapoor, M.Sc., Ph.D.

Former Faculty-University of Alabama at Birmingham, Birmingham, AL 35294


Vectors

Objectives

Physics deals with parameters that can be completely described by a number and are known as “scalars”. Temperature and mass are such parameters.

There are many other parameters that require both size (magnitude) and direction and are known as “vectors” . Examples of vectors are displacement, velocity and acceleration.

In this chapter we will learn:

What is a vector

How to draw a vector.

How to write a vector.

When two vectors are equal.

When two vectors are different.

Geometric vector addition and subtraction

Resolving a vector into its components

The notion of a unit vector

Add and subtract vectors by components

Multiplication of a vector by a scalar

The scalar (dot) product of two vectors

The vector (cross) product of two vectors

What is a Vector?

A vector is a quantity that has magnitude and direction.

Simplest example of a vector is the displacement vector which describes the change in position of an object as it moves from one point (start) to other (finish).

Click play and watch displacement of red disk from origin to final position.

How the vector is drawn? (Select show vector)

It is drawn by an arrow (blue arrow) that points from start point to finish point.  

The length of the arrow (1.13 m) is proportional to the displacement magnitude.

The direction (45° north of east) of the arrow indicated the displacement direction.

Velocity, acceleration etc. are other examples of vector.

How to write a vector?

In books vectors are written in two ways :

Method 1 :  Vectors_2.gif (using an arrow above)

Method 2 : a (using bold face print )

The magnitude of the vector is indicated by italic print : a

When two vectors are same or equal?

Shift any of the two vectors Vectors_3.gif and Vectors_4.gif (Click hold and drag).

A vector can be shifted without changing its length (magnitude) and direction.

The vectors Vectors_6.gif and Vectors_7.gif are just two shifted vectors with same magnitude and direction, therefore both represent the same vector or we can say

Vectors_8.gif

Any two vectors with same magnitude and direction, are considered as same vector or equal.

When two vectors are different?

Two vectors with same magnitude but different direction shown in figure (a) are considered different vectors.

Vectors_9.gif
(a)
Vectors_10.gif
(b)

Two vectors in same direction but of different magnitude shown in figure (b) are also considered as different vectors.

Checkpoint 1

Which of the following statements, if any, involves a vector?
(a) I walked 2 miles along the beach.
(b) I walked 2 miles due north along the beach.
(c) I jumped off a cliff and hit the water traveling at 17 miles per hour.
(d) I jumped off a cliff and hit the water traveling straight down at 17 miles per hour.
(e) My bank account shows a negative balance of −25 dollars.

Adding Vectors Geometrically

Sum of two vectors is a vector.

Vectors_11.gif

Sketch vector Vectors_12.gif  using an appropriate scale.

Sketch vector Vectors_13.gif using the same scale.

Place the tail of Vectors_14.gif at the tip of Vectors_15.gif.

The vector Vectors_16.gif starts from the tail of Vectors_17.gif and terminates at the tip of Vectors_18.gif.

(Click hold and drag the vectors)

Vector addition is commutative.

Vectors_20.gif

(Click hold and drag the vectors)

Negative Vectors_22.gif of a given vector Vectors_23.gif has the same magnitude as Vectors_24.gif but opposite direction.

Vectors_25.gif

Checkpoint 2

In the following figures, is Vectors_26.gif ?

Components of Vectors

A component of a vector is the projection of the vector on an axis.

The components form the legs of a right triangle whose hypotenuse is the magnitude of the vector.

The process of finding the components of a vector is called resolving the vector.

How to compute component of a vector?

To find the projection of a vector along an axis, we draw perpendicular lines from the two ends of the vector to the axis, as shown.

Vectors_29.gif

The projection of a vector Vectors_30.gif on an x axis is its x component Vectors_31.gif).

The projection of a vector Vectors_32.gif on the y axis is the y component Vectors_33.gif).

We can find the components of Vectors_34.gif geometrically from the right triangle.

If θ is the angle that the vector makes with the positive direction of the x axis, and b is the magnitude of Vectors_35.gif.

Vectors_36.gif

How to compute vector magnitude and direction angle?

The magnitude of the vector Vectors_37.gif in terms of its components Vectors_38.gif and Vectors_39.gif is given as  

Vectors_40.gif

For angle θ we can use inverse trigonometric function

Vectors_41.gif

If θ is positive there can be two values

θ = θ  if Vectors_42.gif and Vectors_43.gif both are positive

Vectors_44.gif  if Vectors_45.gif and Vectors_46.gif both are negative.

If θ is negative there can be two values

Vectors_47.gif  if Vectors_48.gif is negative and Vectors_49.gif is positive.

  Vectors_50.gif  if Vectors_51.gif is positive and Vectors_52.gif is negative.

Unit Vector

A unit vector is a vector that has a magnitude of exactly 1 and points in a particular direction.

It does not have any dimension or units.

Its sole purpose is to point — that is, to specify a direction.

The unit vectors in the positive directions of the x, y, and z axes are labeled Vectors_53.gif, Vectors_54.gif and Vectors_55.gif where the hat ^ is used instead of an overhead arrow as for other vectors

Vectors_56.gif

How to write vector in unit vector notation?

Any vector can be expressed in terms of unit vectors and its components.

A vectors  Vectors_57.gif can be expressed as

Vectors_58.gif

Vectors_59.gif

Vectors_60.gif and Vectors_61.gif are vectors, called the vector components of  Vectors_62.gif .

The quantities Vectors_63.gif and Vectors_64.gif are scalars, called the scalar components.

A vector component along any axis can be written as the product of the magnitude of the component and unit vector along that axis.

Example-1

A car moves with velocity of magnitude 30.00 m/s in 30° north of east, find its components, write it in terms of unit vectors and verify your results.

Vectors_65.gif

Vectors_66.gif

Vectors_67.gif

Vectors_68.gif

Vectors_69.gif

Vectors_70.gif

Vectors_71.gif

Vectors_72.gif

Verification of results

Vectors_73.gif

Vectors_74.gif

Vectors_75.gif

We know angle of a vector is given as

Vectors_76.gif

Vectors_77.gif

Vectors_78.gif

Since both x (or i) and y (or j) components are +ve, therefore vector is in 1st quadrant. Hence 30° value is correct.

Example-2

A car moves with velocity of magnitude 30.00 m/s in 30° north of west, find its components, write it in terms of unit vectors and verify your results.

Vectors_79.gif

Vectors_80.gif

Vectors_81.gif

Vectors_82.gif

Vectors_83.gif

Vectors_84.gif

Vectors_85.gif

Vectors_86.gif

Vectors_87.gif

Verification of results

Vectors_88.gif

Vectors_89.gif

Vectors_90.gif

We know angle of a vector is given as

Vectors_91.gif

Vectors_92.gif

Vectors_93.gif

Since x (or i) component is –ve and y (or j) component is +ve, therefore vector is in 2nd quadrant. Hence correct value is 180° -30°= 150°.

Example-3

A car moves with velocity of magnitude 30.00 m/s in 30° south of west, find its components, write it in terms of unit vectors and verify your results.

Vectors_94.gif

Vectors_95.gif

Vectors_96.gif

Vectors_97.gif

Vectors_98.gif

Vectors_99.gif

Vectors_100.gif

Vectors_101.gif

Vectors_102.gif

Verification of results

Vectors_103.gif

Vectors_104.gif

We know angle of a vector is given as

Vectors_105.gif

Vectors_106.gif

Vectors_107.gif

Since x (or i) component is –ve and y (or j) component is -ve, therefore vector is in 3rd quadrant. Hence correct value is 180° +30° = 210°.

Example-4

A car moves with velocity of magnitude 30.00 m/s in 30° south of east, find its components, write it in terms of unit vectors and verify your results.

Vectors_108.gif

Vectors_109.gif

Vectors_110.gif

Vectors_111.gif

Vectors_112.gif

Vectors_113.gif

Vectors_114.gif

Vectors_115.gif

Vectors_116.gif

Verification of results

Vectors_117.gif

Vectors_118.gif

We know angle of a vector is given as

Vectors_119.gif

Vectors_120.gif

Vectors_121.gif

Since x (or i) component is +ve and y (or j) component is -ve, therefore vector is in 4th quadrant. Hence correct value is 360° - 30° = 330°.

Component method of vector addition.

Suppose vector Vectors_122.gif is the sum of two vectors Vectors_123.gif and Vectors_124.gif.

Vectors_125.gif

First determine the x and y components of Vectors_126.gif and Vectors_127.gif, relative to a conveniently chosen x, y coordinate system. Be sure to take into account the direction of the components.

Algebraic sum of x components Vectors_128.gif and Vectors_129.gif) of vector Vectors_130.gif and Vectors_131.gif is the x component Vectors_132.gif of sum vector Vectors_133.gif.

Vectors_134.gif

Similarly, the algebraic sum of y components  Vectors_135.gif and Vectors_136.gif) of vector Vectors_137.gif and Vectors_138.gif is the y component Vectors_139.gif of sum vector Vectors_140.gif.

Vectors_141.gif

Once we know x, y components of Vectors_142.gif, we can use Pythagorean theorem to determine the magnitude |S| of Vectors_143.gif.

Vectors_144.gif

Use inverse tangent function to find the angle that specifies the direction of the resultant vector Vectors_145.gif.

Vectors_146.gif

Example-5

Find the vector sum of the three vectors shown in adjacent figure.

Vectors_147.gif

Vectors_148.gif

Vectors_149.gif

Vectors_150.gif

x - component of is obtained by adding x components of all the three vectors.

Vectors_151.gif

Vectors_152.gif

Vectors_153.gif

y - component of is obtained by adding y components of all the three vectors.

Vectors_154.gif

Vectors_155.gif

Vectors_156.gif

Vector Vectors_157.gif is written as.

Vectors_158.gif

Magnitude is computed as

Vectors_159.gif

Vectors_160.gif

Vectors_161.gif

Vectors_162.gif

Angle is computed as

Vectors_163.gif

Vectors_164.gif

Vectors_165.gif

Since x (or Vectors_166.gif) component is +ve and y (or Vectors_167.gif) component is -ve, therefore vector is in IVth quadrant. Hence correct value is 360° - 51° = 309.

Vector in three dimension

A vector Vectors_168.gif (blue vector) in three dimension has three components and is written as

Vectors_169.gif

Vectors_170.gif

Vectors and the laws of physics

If Vectors_171.gif is a vector, what happens if we rotate or translate the coordinate system?

(Click hold and drag for translation)

Note : Here we are rotating the coordinate system, not the vector.

What we learn?

Vectors do not depend on the location of the origin or on the orientation of the axes.

The components of vector change with rotation of axis.

Which component pair represents the vector?

All the component pairs represent same vector because the magnitude and direction of vector does not change.

Components of a vector are dependent on choice of coordinate system but vector is independent of choice of coordinate system.

All relations of physics are independent of the choice of coordinate system.

Multiplying Vectors

Multiplying a Vector by a Scalar

Multiplication of vector Vectors_173.gif by a scalar s results in a new vector Vectors_174.gif.

Vectors_175.gif

In the component form multiplication is given as

Vectors_176.gif

OR

Vectors_177.gif

The magnitude of new vector is given by.

Vectors_178.gif

We can also get the magnitude of Vectors_179.gif from the component method.

Vectors_180.gif

If s>0 vector Vectors_181.gif has the same direction as vector Vectors_182.gif.

If s<0 vector Vectors_183.gif has a direction opposite to that of vector Vectors_184.gif.

Multiplying a vector by a vector

There are two ways to multiply a vector by a vector:

The Scalar Product

The Vector Product

The scalar product

Also known as "Dot" product.

The scalar product Vectors_185.gif of two vectors Vectors_186.gif and Vectors_187.gif is a scalar and is given by

Vectors_188.gif

where φ is the shorter angle between the two vectors.

Vectors_189.gif

Commutative law applies to a scalar product

Vectors_190.gif

Dot product in unit vector notation

In unit vector notation dot product is written as

Vectors_191.gif

Vectors_192.gif

This product has nine terms out of which magnitude of six terms will be zero because

Vectors_193.gif

Only three terms will be non-zero because

Vectors_194.gif

Therefore

Vectors_195.gif

The vector product

Also known as “cross” product.

The vector product Vectors_196.gif of two vectors Vectors_197.gif and Vectors_198.gif is a vector.

The magnitude of vector Vectors_199.gif is given by

Vectors_200.gif

where φ is the shorter angle between the two vectors.

The direction of Vectors_201.gif is perpendicular to the plane P defined by the vectors  Vectors_202.gif and Vectors_203.gif .

Direction of product vector is given by the right hand rule :

Place the vector and tail to tail

Rotate fingers of the right hand along the shortest angle from Vectors_204.gif to Vectors_205.gif.

The direction of thumb will be the direction of cross product vector (Vectors_206.gif).

Vectors_207.gif

Commutative law does not apply to the vector product

Vectors_208.gif

Vectors_209.gif

Vectors_210.gif

Cross product vectors Vectors_211.gif or Vectors_212.gif of are always perpendicular to the plane containing vectors Vectors_213.gif and Vectors_214.gif.

Cross product in unit vector notation

In unit vector notation vector product is written as

Vectors_215.gif

Vectors_216.gif

This product has nine terms out of which magnitude of three terms will be zero because

Vectors_217.gif

Similarly cross product of other six terms will be non zero because

Vectors_218.gif

How to compute direction of cross product of unit vector?

Vectors_219.gif

Look at the adjacent figure.

Cross product of two unit vector will be third unit vector.

If you are moving in clock wise direction the sign will be positive.

Vectors_220.gif

If you are moving in anti-clock wise direction, the sign will be negative.

Vectors_221.gif

Therefore cross product Vectors_222.gif is given as

Vectors_223.gif

Vectors_224.gif

Vectors_225.gif

Cross product computation Method

Vectors_226.gif can also be calculated by taking det of following matrix.

Vectors_227.gif

components are computed as below.

Vectors_228.gif Vectors_229.gif Vectors_230.gif

As per this rule, the components of vector Vectors_231.gif are given as

Vectors_232.gif