Notes: Vectors and Scalars

A particle moving along a straight line can move in only two directions and we can specify

which directions with a plus or negative sign.

For a particle moving in three dimensions; however, a plus sign or minus sign is no longer

enough to indicate a direction, we must use a vector.

What is a vector?

An object that has both magnitude and direction.

Ok, so what is magnitude?

Mathematically, it is the absolute value. Really it is the size or length of something.

If you say your drove 4 km, it doesn’t matter if it was +4 km or – 4 km, the magnitude is 4

km. You drove 4 km.

What is direction?

If on a single axis, we say that from the objects original position to the objects final position,

can be represented by a plus sign or a minus sign.

So what about when we are not talking about 1-D, but two or three dimensions?

We use terminology such as North, South, East, West, Northeast, Northwest, Southeast, and

Southwest.

However, mathematically, it easier to describe it with angles such as 30° above the x-axis.

What are examples of vectors?

Displacement, velocity, acceleration, force

Obviously, not all substance are vectors e.g. not all substances have direction, so what are

substances without a direction called? Scalars.

Scalar – substances without direction.

What are some examples of some substances with only magnitude and no direction?

Temperature, pressure, energy, mass, and time.

Notes: Vectors and Scalars

(COMPONENTS OF A VECTOR)

A component of a vector is the projection of the vector on an axis; that is, the part of

the vector that translates to three-dimensional space.

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

So as in the image, we have a vector on an x-y plane:

We can break it into component by:

Basically, we have a component of the vector 𝑑 on the x-axis that is called 𝑑𝑥, and a

component of it on the y-axis that is called 𝑑𝑦.

Looking at the diagram above, we see that essentially the vector 𝑑 and its

components form a triangle. This allows us to use trigonometry to create some basic

formulas for describing a vector.

Recall that:

Notes: Vectors and Scalars

𝑐𝑜𝑠𝜃 = 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡

ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒

𝑠𝑖𝑛𝜃 = 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒

ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒

𝑡𝑎𝑛𝜃 = 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒

𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡

We can find 𝑑𝑥 (that is, the amount of vector 𝑑 on the x-axis) by using 𝑐𝑜𝑠𝜃

𝑐𝑜𝑠𝜃 = 𝑑𝑥

𝑑

This can be better written as the formula for 𝑑𝑥 as:

𝑑𝑥 = 𝑑𝑐𝑜𝑠𝜃

Similarly, we can find 𝑑𝑦 (that is, the amount of vector 𝑑 on the y-axis) by using 𝑠𝑖𝑛𝜃

𝑠𝑖𝑛𝜃 = 𝑑𝑦

𝑑

This can be better written as the formula for 𝑑𝑦 as:

𝑑𝑦 = 𝑑𝑠𝑖𝑛𝜃

So, the two above highlighted equations will give you the components of a vector.

Notes: Vectors and Scalars

However, what if we know the components of a vector, but not the magnitude of the

vector or direction.

Look back at the graph above, we have a triangle. Our magnitude is just the distance

of the hypotenuse. So what equation can we use to find the hypotenuse? The

Pythagorean theorem. So to find the magnitude of a vector, we use this formula:

𝑑 = √𝑑𝑥2 + 𝑑𝑦

2

As we already know by now, a vector has both magnitude and direction. We just

found the formula for finding the magnitude, now how do we find the direction

(angle)?

We use 𝑡𝑎𝑛𝜃 to find the angle of a vector:

𝑡𝑎𝑛𝜃 = 𝑑𝑦

𝑑𝑥

(UNIT-VECTOR NOTATION)

Unit vector – a vector that has a magnitude of exactly 1 and points in a particular

direction. It lacks both dimension and unit. Its sole purpose is to point – that is, to

specify direction.

The unit vectors in the positive direction of the x, y, and z axes are labeled

𝑖, 𝑗, 𝑎𝑛𝑑 ��. That is:

�� = 𝑥 − 𝑎𝑥𝑖𝑠 𝑜𝑟 𝑥 − 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛.

�� = 𝑦 − 𝑎𝑥𝑖𝑠 𝑜𝑟 𝑦 − 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛.

�� = 𝑧 − 𝑎𝑥𝑖𝑠 𝑜𝑟 𝑧 − 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛.

Notes: Vectors and Scalars

Unit vectors are very useful for expressing other vectors for instance:

Where i and j are the vector components, and 𝑎𝑥 and 𝑎𝑦 are the scalar

components of vector ��.

(ADDING VECTORS BY COMPONENTS)

To add vectors, we can add their components axis by axis. Take

𝑟 = �� + ��

We can rewrite this as:

𝑟 = (𝑎𝑥 + ��𝑦 + 𝑎𝑧) + (𝑏𝑥 + ��𝑦 + 𝑏𝑧

)

Notes: Vectors and Scalars

𝑟 = (𝑎𝑥 + ��𝑥)�� + (𝑎𝑦 + ��𝑦)�� + (𝑎𝑧 + 𝑏𝑧 )��

𝑟 = 𝑟𝑥 �� + 𝑟𝑦 �� + 𝑟𝑧 ��

Where,

𝑟𝑥 = 𝑎𝑥 + 𝑏𝑥

𝑟𝑦 = 𝑎𝑦 + 𝑏𝑦

𝑟𝑧 = 𝑎𝑧 + 𝑏𝑧

Product of Vectors:

- Let ��, ��, and �� be unit vectors in the x, y, and z directions. Then:

�� ∙ �� = �� ∙ �� = �� ∙ �� = 1 and �� ∙ �� = �� ∙ �� = �� ∙ �� = 0

Meaning that only vectors in the same direction can be multiplied together

whereas if they are in opposite direction than they cancel out.

(ADDING VECTORS GEOMETRICALLY)

Resultant vector (or vector sum) – the sum of two vectors that create another vector.

To add two-dimensional vectors together geometrically:

(1) Sketch a vector �� to some convenient scale and at the proper angle.

(2) Sketch a vector �� to the same scale, with its tail at the head of vectors ��, again at

the proper angle

(3) The vector sum 𝑠 is the vector that extends from the tail of �� to the head of ��.

Notes: Vectors and Scalars

Many of the same laws that you encountered in algebra, apply in the same way to

vectors.

Commutative law – states that the order of addition does not matter. Adding vector

�� to vector �� gives to same result as adding vector �� to vector ��.

Associative law – is an extension of the commutative law to more than two vectors

by stating that the order of addition does not matter for vectors ��, ��, and 𝑐.

Vector Subtraction – states that if you have a vector −�� than it has the same

magnitude as vector �� except it is in the opposite direction.

NOTE: Although we have used displacement vectors here, the rules for addition

and subtraction hold for vectors of all kinds, whether they represent velocities,

accelerations, or any other vector quantity.

(MULTIPLYING VECTORS)

Three ways in which vectors can be multiplied:

1. Multiplying a Vector by a Scalar

2. Dot Product (Multiplying a Vector by a Vector)

3. Cross Product (Multiplying a Vector by a Vector)

(1) Multiplying a Vector by a Scalar:

If we multiply a vector �� by a scalar s, we get a new vector. The new vectors

magnitude is the product of the magnitude of vector �� and the absolute value of

Notes: Vectors and Scalars

scalar s. The new vectors direction is the direction of vector �� if scalar s is positive

unless scalar s is negative then the new vectors direction is negative.

Let s be the scalar and let �� be a vector. Let 𝑑 be the vector when the scalar and the

vector �� are multiplied together.

𝑑 = 𝑠��

(2) The Scalar Product (Dot Product):

The Scalar Product that is most often called the dot product is one of the ways to

multiply a vector by another vector which is to say to multiply a quantity that has a

magnitude and a direction (��)by another quantity that has a magnitude and a

direction ( ��).

Where, 𝜃 is the angle between the two vectors.

IMPORTANT NOTE: There are actually two such angles when two vectors are

multiplied together.

θ and (360°-θ)

Either angle can be used in the dot product because their cosines are the same

but this is not true for the cross product or anytime you have the sine of

something.

Dot Product produces a scalar product and not another vector i.e. when two vectors

or multiplied together by the dot product, they produce a quantity with only a

magnitude and not a direction.

A dot product can be regarded as the product of two quantities:

1. The magnitude of one of the vectors.

2. The scalar component of the second vector along the first vector.

Notes: Vectors and Scalars

(3) The Vector Product (Cross Product):

The Vector Product is commonly called the Cross Product. It is when you multiply

two vectors together to produce another vector. Note that the word multiply is used

loosely because it is different than how you typically accustomed to multiplying.

Notes: Vectors and Scalars

Essentially it is when a vector (��) encounters another vector (��) to create a new

vector (𝑐) that has both a magnitude and a direction. That is:

𝑐 = �� x �� = 𝑎𝑏𝑠𝑖𝑛∅

𝑐 = �� x �� = |

�� �� ��𝑎𝑥 𝑎𝑦 𝑎𝑧

𝑏𝑥 𝑏𝑦 𝑏𝑧

| = |

�� �� ��𝑎𝑥 𝑎𝑦 𝑎𝑧

𝑏𝑥 𝑏𝑦 𝑏𝑧

�� ��𝑎𝑥 𝑎𝑦

𝑏𝑥 𝑏𝑦

|

= (𝑎𝑦𝑏𝑧 − 𝑎𝑧𝑏𝑦)�� + (𝑎𝑧𝑏𝑥 − 𝑎𝑥𝑏𝑧)�� + (𝑎𝑥𝑏𝑦 − 𝑎𝑦𝑏𝑥)��

∅ is the small of the two angles between vector (��) and vector (��); this is because sin∅

and sin(360°-∅) differ in algebraic sign.

The direction of vector 𝑐 (which is the vector created from crossing vector �� and

vector ��) is perpendicular to the plane that contains vector �� and vector ��.

EXAMPLE PROBLEM 1:

A small airplane leaves an airport on an overcast day and is sighted 215 km away, in a

direction making an angle of 22° East of due North.

How far east is the airplane from the airport when sighted?

How far north is the airplane from the airport when sighted?

STEP 1: The first part to solving physics problems and any work problem is to pick out what

relevant and irrelevant data you have in a problem. So ask yourself, what do we know so

far?

We have a number that 215 km, but what is that number? It is the magnitude of the

displacement vector (look at graph). So, we can say that d = 215 km.

Notes: Vectors and Scalars

We also have an angle that is 22° east of due north. What does that mean? Well if you recall

a unit circle where the y-axis is 90° and is pointing straight up, the same direction as North,

we can call that our north direction. Similarly, east would be heading to the right. So

basically the is 22° east of due north says that we start at north and we head east. So we are

22 degrees off the y-axis.

So as you can see, we end up with an angle that is 𝜃 = 90° − 22° = 68°.

Now we break the vector down into its components.

Notes: Vectors and Scalars