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Table of Contents:

1. Standards and Units (Young et al., 2016)
2. Uncertainty and Significant Figures (Young et al., 2016)
   1. Uncertainty
   2. Significant Figures
      1. Multiplication or Division
      2. Addition or Subtraction
      3. Precision vs. Accuracy
3. Vectors (Young et al., 2016)
   1. Vector Notations
   2. Representation of a Vector
   3. Parallel vs. Antiparallel vs. Equal Vectors vs. Negative of a Vector
   4. Vector Addition and Subtraction
      1. Vector Addition
      2. Vector Subtraction
   5. Multiplying a Vector by a Scalar
   6. Components of a Vector
      1. The Angle
      2. How can you use Components to perform Vector Calculations?
         1. Finding Magnitude and Direction
         2. Multiply a vector by a scalar
         3. Vector Sum
   7. Unit Vectors
   8. Product of Vectors
      1. Scalar Product
         1. Projection of a Vector
         2. The Angle
      2. Vector Product
         1. The Angle
   9. Summary for Vector Addition
4. Common Errors in Understanding (Young et al., 2016)

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Standards and Units (Young et al., 2016)

The International System (SI) outlines the standard units of measurement³, which are universal. Fundamental quantities like time, length or mass can only be defined by stating how to measure them. Quantities like speed can then be defined by utilizing two or more of these fundamental quantities.

• A second is defined by the atomic clock. When cesium-133 atom is bombarded with microwave radiation of a specific frequency, the cesium atom transitions from one energy state to another.
  • The two energy states refer to the reversal of spin of the outermost electron in the cesium atom.
  • Specific frequency depends on the energy difference between the two states and is determined to be 9,192,631,770 cycles per second; .
  • One second is then defined as the time required by the microwave radiation to complete 9,192,631,770 cycles.
• A meter is the distance in vacuum travelled by light in 1/299,792,458 second.
  • Speed of light in vacuum is measured to be 299,792,458 m/s.
  • Light travels 299,792,458 meters in 1 second. This means that it travels 1 meter in 1/299,792,458 second.
• A kilogram is defined by the value of Planck’s constant (NIST, 2020), h = 6.62607015 ×10⁻³⁴ J/s = 6.62607015 ×10⁻³⁴ kg m² s^-1 .
  • m is defined by speed of light and s is defined by the frequency of microwave radiation needed for ¹³³Cs transition (as stated above).

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Uncertainty and Significant Figures (Young et al., 2016)

Uncertainty

When we take measurements, the measurement is reliable only to the nearest possible value that the measuring device can measure. For example, measurements made using a centimeter scale are reliable up to the nearest centimeter. If the scale yields a result of 15.0 cm, it is possible that the actual length is 14.8 cm or 15.1 cm.

Uncertainty (also referred to as the error) is defined as the maximum possible difference between the measured value and the true value. Smaller the uncertainty, larger is the accuracy of the measured value.

Uncertainty	25.0+/-0.3 m	The true value lies somewhere between 24.7 m and 25.3 m.
Fractional error/uncertainty	0.3/25.0 = 0.012	Uncertainty divided by the measured value.
Percent error/uncertainty	0.012*100% = 1.2%	A scale that measures 25.0 m +/- 1.2% means that the true length differs from 25.0 m by at most 1.2%.

Significant Figures

Sometimes, measurements do not include uncertainties and uncertainty is implicitly derived from the number of significant digits listed.

Measurement	Significant Digits	Uncertainty
2.85 m	3	~0.01 m
158 km	3	~1 km4
0.59 m	25	~0.01 m
0.590 m	3	~0.001 m
5.00 cm	36	~0.01 cm
5900 km	2 or 3 or 4	~100 km 0r ~10 km or ~1 km7
4800.056 km	78	~0.001 km
9.25 x 10-4 m/s	3	~0.000001 m/s or 0.01 x 10-4 m/s

Method: In 2.85 m, there are 3 significant digits. The last one is in the hundredths place and so the uncertainty is ~0.01 m.

Multiplication or Division

When two numbers are multiplied or divided, the resulting value can not have significant figures that are more than the factor with the least number of significant digits.

For example: 5.9678 x 0.43 = 2.6 (Max of two significant figures as highlighted)

Addition or Subtraction

When two numbers are added or subtracted, uncertainty in the resulting value is given by the number with the largest uncertainty (or least number of digits after the decimal point).

For example: 4.876 + 3.2 + 0.25 = 8.3 (Only one digit after the decimal allowed)

Precision vs. Accuracy

A measurement can have 8 significant figures making it very precise but the result can still be inaccurate if the equipment is faulty. A measurement can have two significant figures, making it not so precise but the result can be very close to the true value making it highly accurate. The goal of measurements is to be both accurate and precise.

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Vectors (Young et al., 2016)

A vector is a physical quantity that has both a magnitude and a direction. It is different from a scalar quantity, which only has a magnitude but no direction associated with it.

Examples of scalar quantities: distance, time, temperature, mass, density, energy, speed, voltage, current, etc.

Examples of vector quantities: velocity, force, displacement, acceleration, momentum, electromagnetic field, etc.

Distance is a scalar quantity because it simply tells you how far an object moved. On the other hand, displacement is defined as the change in an object’s position, which includes how far the object moved and in what direction. Hence, displacement is a vector quantity.

Vector Notations

A vector is denoted by placing an arrow on top of the letter or symbol as shown below:

The magnitude of a vector (scalar quantity, always positive) is represented as follows:

Representation of a Vector

A vector is represented using an arrowhead with the tail at the starting point and the tip at the final point. The length of the arrow is representative of the magnitude of the given vector and the direction of the arrow aligns with the direction of the given vector.

In Figure 2, displacement vector is shown by the blue arrow, a straight line from point A to B. Displacement vector is independent of the path taken by the object. In this case, distance travelled (green dotted line) is much greater than the magnitude of the displacement vector. If the object started at point A and subsequently returned to point A, the displacement for the whole trip will be zero but distance will be a non-zero quantity.

Parallel vs. Antiparallel vs. Equal Vectors vs. Negative of a Vector

Parallel Vectors	Antiparallel Vectors	Equal Vectors	Negative of a Vector
Two vectors are said to be parallel, if they have the same direction	Two vectors are said to be antiparallel, if they have opposite direction	Two vectors are said to be equal, if they have the same magnitude and are in the same direction	When two vectors have the same magnitude but opposite direction
Vectors can have different magnitudes	Vectors can have different magnitudes	Vectors can have different spatial location	Vectors can have different spatial location

Vector Addition and Subtraction

Vector Addition

What does it mean to add two vectors, say displacementand displacement ?

Let’s say an object moves a certain length along a certain direction and the displacement is given by, . It then goes under a second displacement given by . The resulting displacement,, is the same as if the object had started at the initial point of and stopped at the final position of . In other words, is the vector sum or resultant vector of vectors and (See Error #1).

= + ……(1)

Vectors can be added using three methods as illustrated below:

Method 1:

This is the most used method that reduces the likelihood of making errors.

Method 2:

Method 3:

What if you want to add more than two vectors, say , , & ?

There are two methods with which you can add multiple vectors.

Method 1:

Method 2:

Vector Subtraction

Vectors can be subtracted using two methods.

Method 1:

Think of vector subtraction as vector addition. Instead of subtracting from , add –to . In other words,

– = + (-)……(5)

This way, the rules of vector addition apply. First, determine the negative of and then add it to as shown in Figure 9.

This method reduces confusion and the likelihood of making an error.

Method 2:

Multiplying a Vector by a Scalar

Vectors can be multiplied by scalar quantities as follows:

= k…..(6)

• The value of scalar, k, is multiplied by the magnitude of to get the magnitude of .
  • If k > 1, increases. For example, = 2means that is twice as big as .
  • If 0 <k < 1, decreases. For example, = 0.5means that is half of .
• The sign of scalar k, gives information regarding the direction of the vector, .
  • If k is positive, is in the same direction as .
  • If k is negative, is in the opposite direction as .

Components of a Vector

Instead of using vector diagrams, vectors can be easily added using the components of the vector given.

In the cartesian coordinate system, a vector makes an angle with respect to some reference direction, usually taken to be the positive x-axis. In Figure 11, makes an angle with the positive x-direction.

The components of a vector can be calculated from the magnitude of and the angle it makes with the reference direction, in this case, positive x-axis, which is given by .

The Angle

It is important to note the properties of the angle in consideration, .

• is positive for counter-clockwise angles, i.e., starting at positive x-axis and going towards positive y-axis.
• is negative for clockwise angles, i.e., starting at positive x-axis and going towards negative x-axis.

The Angle in Four Quadrants

The angle that the vector makes with the reference direction can be measured in the counter-clockwise direction or clockwise direction. An angle measured in the counter-clockwise direction will yield a positive result while an angle measured in the clockwise direction will yield a negative result.

In the first quadrant, +51° can also be stated as -309°.

In the second quadrant, +140° can also be stated as -220°.

In the third quadrant, -144° can also be stated as +216°.

In the fourth quadrant, -31° can also be stated as +329°.

With the magnitude of and angle known, the components can be easily determined by applying Pythagoras theorem to Figure 11:

cos = A_x/ and sin = A_y/…..(7)

A_x = cos and A_y = sin¹⁰……(8)

Quadrant	Sin and Cos	Ax and Ay
I (0°<<90°)	Cos = +ve Sin = +ve	Ax = +ve Ay = +ve
II(90°<<180°)	Cos = -ve Sin = +ve	Ax = -ve Ay = +ve
III(180°<<270°)	Cos = -ve Sin = -ve	Ax = -ve Ay = -ve
IV(270°<<360°)	Cos = +ve Sin = -ve	Ax = +ve Ay = -ve

Special Cases:

Angle	Sin and Cos	Ax and Ay
= 0°/360°	Cos = +ve Sin = 0	Ax = +ve Ay = 0
= 90°/-270°	Cos = 0 Sin = +ve	Ax = 0 Ay = +ve
= 180°/-180°	Cos = -ve Sin = 0	Ax = -ve Ay = 0
= 270°/-90°	Cos = 0 Sin = -ve	Ax = 0 Ay = -ve

How can you use Components to perform Vector Calculations?

Finding Magnitude and Direction

Components of a vector can be used to calculate the vector’s magnitude and direction. Given A_x and A_y, the magnitude of is given by applying the Pythagorean theorem¹¹:

= …..(9)

The direction of , given by (measured from positive x-axis and towards positive y-axis), is calculated as follows (See Error#3):

……(10)

…….(11)

Note: Equations 9, 10 and 11 are applicable only if x and y axes are perpendicular to each other.

Multiply a vector by a scalar

When a vector is multiplied by a scalar as shown in equation 6,

= k

Each component of is k times the corresponding component of ,

B_x = kA_x and B_y = kA_y

Proof:

which is true as discussed earlier.

Vector Sum

For a given vector sum, = + ++,

where , ,&are n-dimensional vectors;

R_x = A_x + B_x + C_x + D_x

R_y = A_y + B_y + C_y + D_y

R_z = A_z + B_z + C_z + D_z

and so on, such that,

R_n = A_n + B_n +C_n +D_n….(12)

This method can be replicated for a sum of N-vectors, in which case,

R_x = A_x + B_x + C_x + D_x + E_x………………..+ N_x

R_y = A_y + B_y + C_y + D_y + E_y………………..+ N_y

R_z = A_z + B_z + C_z + D_z + E_z………………..+ N_z

and so on, such that,

R_n = A_n + B_n +C_n +D_n + E_n………………..+ N_n….(13)

Vector sum using components can be easily visualized in 2D Cartesian coordinate system.

Unit Vectors

A unit vector is defined as a unitless¹³ quantity with a magnitude that equals one and it points towards a certain direction.

Unit Vector Notation: î (letter with a hat/caret on top)

Vectors can then be expressed using components and unit vector notations.

= A_x + A_y + A_z

= B_x + B_y + B_z

Vector sum is then given by:

= +

= A_x + A_y + A_z + B_x + B_y + B_z

= (A_x + B_x) + (A_y + B_y) + (A_z + B_z)……(14)

= R_x + R_y + R_z……..(15)

Notice in equation 14 that x components of the two vectors can be grouped together and subsequently added. The same can be done for y and z components of the two vectors. Once added, the final values are equivalent to x, y and z component of the resultant vector (equation 15).

Product of Vectors

Scalar Product

Scalar product is also called dot product because it is represented using the following notation:

It is called a scalar product because and are vectors but is a scalar quantity.

Scalar product can be calculated using two methods.

Method 1:

…….(16)

Take two vectors, in this case and , such that they are placed tail to tail and the angle between them, denoted by , lies between 0° and 180°. Dot product of these two vectors is then equal to the magnitude of multiplied by the projection of onto . Alternatively, it is equal to the magnitude of multiplied by the projection of onto .

……(17)

Thus, scalar product is commutative, because .

Projection of a Vector

A vector can be projected onto a certain direction.

Alternatively,

Note:

The Angle

As stated earlier, can range between 0° and 180°.

Important: The dot product of two perpendicular vectors is always equal to zero.

Method 2:

Scalar product can also be computed using vector components. For two given vectors,

= A_x + A_y + A_z

= B_x + B_y + B_z

= (A_x + A_y + A_z ) • (B_x + B_y + B_z)

= A_xB_x(•) + A_xB_y(•) + A_xB_z(•) + A_yB_x(•) + A_yB_y(•) + A_yB_z(•) + A_zB_x(•) + A_zB_y(•) + A_zB_z(•)……(18)

Using equation 16,

•= •= •= (1)(1)(cos0°) = 1

•= •=•=•=•=•=(1)(1)(cos90°) = 0

Substituting in equation 18,

= A_xB_x + A_yB_y + A_zB_z……(19)

The dot product of two vectors, and , is equal to the sum of the product of their components in a given direction.

Vector Product

Vector product is also called cross product because it is represented using the following notation:

It is called a vector product because the result of is a vector quantity.

Vector product can be calculated using two methods.

Method 1:

Let two vectors, say and lie in a plane with an angle between them. The magnitude of the cross product between these two vectors can be calculated as follows:

…..(20)

The magnitude of the cross product is equal to the magnitude of multiplied by the component of perpendicular to . Alternatively, the magnitude of cross product is equal to the magnitude of multiplied by the component of perpendicular to .

The magnitude of a cross product is commutative;

The direction of is given by the right-hand rule and is always perpendicular to the plane in which the two vectors lie. If and lie along x and y axes respectively, then will lie along the z-axis. The right-hand rule tells you whether the direction is along the positive z-axis or negative z-axis. Place your hand in the direction of the first vector (as shown in figure 35), in this case , and curl your fingers towards the second vector, in this case . The thumb will point in the direction of . Notice that for , when you place your hand along and curl your fingers towards , the thumb points in the opposite direction. This means that and are equal in magnitude and opposite in direction. Thus, vector product is anticommutative; = –.

The Angle

Special Case:

If = 0° or 180°, sin is equal to zero and therefore, is also zero. In other words, cross product between two parallel or antiparallel vectors is always zero. Consequently, cross product of a vector with itself is always zero.

Method 2:

Cross product can also be computed using vector components. For two given vectors,

= A_x + A_y + A_z

= B_x + B_y + B_z

= (A_x + A_y + A_z ) (B_x + B_y + B_z)

Cross products can be easily carried out using matrices, especially for vectors in 3D.

Draw a matrix with unit vectors in the first row, components of in the second row and components of in the third row. At the top of the unit vectors, start with the positive sign and subsequently alternate the sign between positive and negative, i.e., +, -, +, -, +, -, and so on. The second matrix provides labels for the each position in a 3×3 matrix. The general rule for matrix multiplication is as follows: Start with the first sign (+) and hold the value at C11 constant while the rest of the values in the first row and first column are ignored, in this case, C12, C13, C21 and C31. Next, the remaining four values are cross multiplied, such that C22 is multiplied by C33 and C23 is multiplied by C32 making sure a negative sign is added to the result of the latter multiplication. These steps are repeated for values at C12 and C13. For C12, make sure you start with the sign on top which would be a negative.

Applying these steps to matrix:

= +(A_yB_z – A_zB_y) – (A_xB_z – A_zB_x) + (A_xB_y – A_yB_x), which can also be written as;

= +(A_yB_z – A_zB_y) +(A_zB_x – A_xB_z ) + (A_xB_y – A_yB_x)…..(21)

If the cross product between and is equal to , then the x-component of , C_x = A_yB_z – A_zB_y; the y-component of , C_y = A_zB_x – A_xB_z; and the z-component of , C_z = A_xB_y – A_yB_x.

Alternatively,

= (A_x + A_y + A_z ) (B_x + B_y + B_z)

= A_xB_x()+ A_xB_y() + A_xB_z( )+ A_yB_x( )+ A_yB_y ()+ A_yB_z( )+ A_zB_x( ) + A_zB_y( ) + A_zB_z( )……(22)

Using equation 20 and right-hand rule:

== = (1)(1)(sin0°) = 0

= , = –, = –, = , = , = –

Substituting these values into equation 22,

= A_xB_y() + A_xB_z( –)+ A_yB_x( –)+ A_yB_z( )+ A_zB_x() + A_zB_y( – )

Bringing like terms together,

= A_yB_z( ) + A_zB_y( – ) + A_xB_z( –) + A_zB_x() + A_xB_y() + A_yB_x( –)

= (A_yB_z – A_zB_y) + (A_zB_x – A_xB_z) + (A_xB_y – A_yB_x)….(23)

which is the same as equation 21.

Summary for Vector Addition

Vectors can be added in two ways:

• Using diagrams
  • Vectors can be added using diagrams but this method is not practical when dealing with vectors in three dimensions or more. For two vectors, scale diagrams can yield correct results. For example, you walk 400 km due North followed by 300 km due East. You can scale down 400 km to 4 units and 300 km to 3 units so that it can fit on a graph paper. In other words, the new values are scaled down by a factor of 100. When you measure the resultant displacement vector, it will be equal to 5 units which can be scaled up by 100 units yielding a displacement of 500 km (See Figure 36 below).
• Using vector components
  • When components of vectors are given, they can be used to calculate the vector sum. The magnitude and direction of the resultant vector can be calculated using equations 9 and 10.

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Common Errors in Understanding (Young et al., 2016)

Error #1:

When adding two vectors, such that, = + , it is a common mistake to think that the magnitudes add as scalar quantities;

= +

This is wrong. does not necessarily equal the sum of and . Instead, depends on , and the angle between and . In a special case where and parallel, magnitude of does equal magnitude ofplus magnitude of . However, if and are antiparallel, = | – |¹⁵.

Back to Vector Addition

Error #2:

Components of a vector are not vectors. They are signed scalars. If A_x is positive, it lies along the positive x-direction. If A_x is negative, it lies along the negative x-direction. Similarly, if A_y is positive, it lies along the positive y-direction, and if A_y is negative, it lies along the negative y-direction.

Back to Vector Components

Error #3:

Any tangent value will yield two angles between 0° and 360°, for instance, tan is -1 at = 135° and = 315°. Similarly, tan = 0.4244 for both = 23° and = 23° + 180° = 203°. In general, angles separated by 180° will have the same tangent value. How does one find out which angle is the correct choice? The correct value of can be identified by analyzing the given vector components. For tan = 0.4244, if A_x and A_y are both positive (1st quadrant), the correct value of will be 23° and if A_x and A_y are both negative (3rd quadrant), the correct value of will be 203°.

Back to Magnitude and Direction

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1. Young, H.D. et al. (2016) Sears and Zemansky’s university physics: With modern physics. 14th edn. Boston: Pearson. ↩︎
2. Definitions of SI base units (2020) NIST. Available at: https://www.nist.gov/si-redefinition/definitions-si-base-units (Accessed: 24 December 2024) ↩︎
3. The adopted standards for different units have a long history, and have changed over time with newer standards being more precise. For example, until 2018, 1 kg was defined as the mass of a cylinder made of platinum-iridium alloy locked in a vault in Paris. ↩︎
4. Both 2.85m and 158 km have 3 significant digits but the uncertainties are vastly different.  ↩︎
5. Zeroes to the left of non-zero digits do not count. ↩︎
6. Zeroes after a decimal point are significant. ↩︎
7. Numbers like 5900 should be stated using scientific notation, as 5.9×10³ or 5.90×10³ . This makes it easier to determine the uncertainty, for example, 5.9×10³ has two significant digits and so the uncertainty is ~100 and 5.90×10³ has three significant digits and the uncertainty is ~10. ↩︎
8. Zeroes between non-zero numbers are significant. ↩︎
9. 2D Cartesian coordinate system is described by an origin and two axes labelled x and y. The position of any point on the plane can then be given by two values, an x-coordinate, which tells you how far the point is from the origin along the x-axis, and a y-coordinate, which tells you far the point is from the origin along the y-axis. ↩︎
10. For the equations to be valid, must be measured counter-clockwise, i.e., from positive x-axis towards positive y-axis. If the angle is given from a different reference direction or is measured clockwise, first determine what the angle would be when you start at +x-axis and go towards +ve y-axis, then apply these equations to find the components. ↩︎
11. Disregard the negative root because magnitude is a scalar quantity. ↩︎
12. Since makes an angle, , with the reference direction (+ve x-axis) such that 90°<<180°, this means that the x component of the vector, i.e., B_x, will be negative. ↩︎
13. Unit vectors have no units. ↩︎
14. The projection can be calculated by applying pythagorean theorem to this right angle triangle. ↩︎
15. The difference in magnitudes is placed inside an absolute symbol because the magnitude of a vector is a scalar quantity which is always positive. ↩︎