Direction of a Vector

The direction of a vector is the angle made by the vector with the horizontal axis, that is, the X-axis. The direction of a vector is given by the counterclockwise rotation of the angle of the vector about its tail due east. For example, a vector with a direction of 45 degrees is a vector that has been rotated 45 degrees in a counterclockwise direction relative to due east. Another convention to express the direction of a vector is as an angle of rotation of the vector about its tail from east, west, north or south. For example, if the direction of a vector is 60 degrees North of West, it implies that the vector pointing West has been rotated 60 degrees towards the northern direction.

The direction along which a vector act is defined as the direction of a vector. Let us learn the direction of a vector formula and how to determine the direction of a vector in different quadrants along with a few solved examples.

What is the Direction of a Vector?

The direction of a vector is the orientation of the vector, that is, the angle it makes with the x-axis. A vector is drawn by a line with an arrow on the top and a fixed point at the other end. The direction in which the arrowhead of the vector is directed gives the direction of the vector. For example, velocity is a vector. It gives the magnitude at which the object is moving along with the direction towards which the object is moving. Similarly, the direction in which a force is applied is given by the force

Direction of a Vector Formula

The direction of a vector formula is related to the slope of a line. We know that the slope of a line that passes through the origin and a point (x, y) is y/x. We also know that if θ is the angle made by this line, then its slope is tan θ, i.e., tan θ = y/x. Hence, θ = tan^-1 (y/x). Thus, the direction of a vector (x, y) is found using the formula tan^-1 (y/x) but while calculating this angle, the quadrant in which (x, y) lies also should be considered.

Steps to find the direction of a vector (x, y):

• Find α using α = tan^-1 |y/x|.
• Find the direction of the vector θ using the following rules depending on which quadrant (x, y) lies in:

Quadrant in which (x, y) lies	θ (in degrees)
1	α
2	180° – α
3	180° + α
4	360° – α

To find the direction of a vector whose endpoints are given by the position vectors (x₁, y₁) and (x₂, y₂), then to find its direction:

• Find the vector (x, y) using the formula (x, y) = (x₂ – x₁, y₂ – y₁)
• Find α and θ just as explained earlier.

Let us now go through some examples to understand how to find the direction of a vector.

How to Find the Direction of a Vector?

Now that we know the formulas to determine the direction of a vector in different quadrants, let us go through an example to understand the application of the formula.

Example 1: Determine the direction of the vector with initial point P = (1, 4) and Q = (3, 9).

To determine the direction of the vector PQ, let us first determine the coordinates of the vector PQ

(x, y) = (3-1, 9-4) = (2, 5). The direction of the vector is given by the formula,

θ = tan^-1 |5/2|

= 68.2° [Because (2, 5) lies in the first quadrant]

The direction of the vector is given by 68.2°.

Example 2: Consider the image given below.

The vector in the above image makes an angle of 50° in the counterclockwise direction with the east. Hence, the direction of the vector is 50° from the east.

Important Notes on Direction of a Vector

• The direction of a vector can be expressed by the angle its tail forms with east, north, west or south.
• After determining the value of tan^-1 |y/x|, we can apply the respective formula for each quadrant.
• The direction of a vector is can also be given by the angle made by the vector in the counterclockwise direction from east.

Direction of a Vector Examples

What is the Direction of a Vector Formula?

How to Find the Direction of a Vector?

What does the Direction of a Vector Represent?

How Do you Find the Direction of a Vector Given its Components?

The Direction Of A Vector – Explanation and Examples

In the realm of vector geometry, the direction of a vector plays a fundamental role. The direction of  a vector  is defined as:

“The direction of a vector is the direction along which it acts.”

Keeping the importance of direction in mind, let’s move forward.   

We will be covering the following topics in this section:

• What is the direction of a vector?
• How to find the direction of a vector?
• What is the formula for finding the direction of a vector?
• Examples
• Practice problems 

What Is The Direction Of A Vector?

A vector is a physical quantity described by a magnitude and direction. A vector quantity is represented by a vector diagram and hence has a direction—the orientation at which the vector points is specified as the direction of a vector. 

In convention, where its vector diagram represents a vector, its direction is determined by the counterclockwise angle it makes with the positive x-axis. According to a scale, the vector diagram is a line with an arrowhead that denotes the direction of the vector. 

A = |A| Â

|A| represents magnitude, and  represents the unit vector.

For example, to describe a body’s velocity completely, we will have to mention its magnitude and direction. This means that we will have to mention how fast it is going in terms of distance covered per unit time and describe what direction it is headed. 

So, if we say a car is moving at 40 km/hr. This statement only describes the speed of the body. If someone says a car is moving at 40 km/hr and is headed North. This statement is describing the velocity of the car. It tells us the magnitude by which the car is moving and the direction in which it is headed.  

This is why, for us to describe a vector, the direction is just as vital and the magnitude. If we were to say that the chocolates are 3 meters outside of the classroom towards the North, it would make more sense. 

We have seen in the above-mentioned example how the direction is important to a vector quantity.

The arrowhead donates the direction of the vector, and the tail represents the point of action. There are two conventional ways to describe the direction of a vector.

• A vector’s direction can be described by the angle its tail forms with East, North, West, or South. For instance, while describing a vector, it can be said that a vector is directed 80° South of East. This means that the vector has been rotated 80° from the East towards the South. The purple vector represents this. 

Similarly, another vector can be 65° South of the West. This means that it is directed 65° about the tail from the West toward the South. The green vector denotes this.

Another way to describe a vector is by the counterclockwise angle of rotation from the due “East.” According to this, a vector with a direction of 50° is directed 50° from the East.

Let’s see this vector diagram. If a vector is said to have a direction of 50°. The trick to figuring it out is to pin down the tail of the vector aligned with the due East or the x-axis. Now rotate the vector 50° counterclockwise about its tail. 

Now take another example. Suppose a vector has a direction of 200°. This means the vector’s tail is pinned down at the East and is then rotated 200° about counterclockwise.

Similarly, a Rectangular coordinate system can also be used. In that case, the angle will be calculated from the positive x-axis. 

Now, let’s consider some examples to understand this concept better. 

Example 1

Draw a vector 30° North of West. 

Solution

Example 2

Draw a vector with direction 60° East of North.

Solution

How To Find The Direction Of A Vector?

The direction of a vector is determined by the angle it makes with the horizontal line. 

There are two methods of finding the direction of a vector: 

1. Graphical Method
2. Using Inverse Tangent Formula

Graphical Method

The graphical method, as the name suggests, requires you to draw the vector graphically and then calculate the angle. The steps for the graphical method are as follows: 

1. Draw the individual vectors with their tails at origin and according to their angles. 
2. Using the head-to-tail rule, add the vectors. 
3. The resultant vector R is directed from the tail of the first vector A to the head of the second vector B. 
4. The magnitude and direction of the vector are then determined by using the ruler and protractor. The length of the resultant vector R will give it magnitude. 
5. For direction, draw a line parallel to the x-axis passing through the starting point of the resultant vector R. Measure the angle between the horizontal line and the resultant.

However, here is the problem: This method is only for basic understanding. It gets complicated if you have to add multiple vectors and does not always give the most accurate result. There is always a chance of human error. Therefore, we have the second method: 

The Inverse Tangent Formula 

We use the inverse tangent function to find the angle it makes with the horizontal line. 

This is possible if you have the initial and final coordinate points of a vector in a plane. It is given by:

θ = tan-1 (y/x)

Example 3

A vector is directed from origin to the (3,5). Determine its direction. 

Solution

Here we can see that, 

a = x = 3

b = y = 5

θ = tan-1 (a/b) 

θ = tan-1 (3/5)

θ = 30.9°

The vector is directed at 30.9° from the x-axis. 

Now, consider a case where the tail is not located at the origin, but rather the vector is placed somewhere else in the plane. In this case, the formula is modified as follows: 

By Pythagorean property, we know: 

tanθ = Δy/Δx

tanθ = (y2 – y1)/(x2 – x1)

θ = tan-1 (y2 – y1)/(x2 – x1)

So, the formula is modified as:

θ = tan-1 (y1 – y0)/(x1 – x0)

The angle given by this is from the horizontal line, running parallel to the x-axis. 

Let’s solve some examples to understand this concept. 

Example 4

Find the direction of the vector located from A(2,1) to B(6,9)

Δx = x1 – x0 = 6 -2 = 4

Δy = y1 – y0 = 9 -1 = 8

Solution

Using formula:

θ = tan-1 (y1 – y0)/(x1 – x0)

θ = tan-1 (8/4)

θ = 63.4°

The Conventions For The Direction Of A Vector

Let’s move on to a much tougher case. 

We have seen that in the above example, the vector lies in the First Quadrant. Let’s see how it works for the rest of the Quadrants. This can be determined by the signs of the vector’s coordinates, which determine the quadrant in which the angle lies. 

For this, certain conventions should be followed:

1. If both the coordinates are positive, then the angle exists in the first quadrant and is considered as the standard angle. θ = Ⲫ
2. If the y-coordinate is positive, but the x-coordinate is negative, then the angle exists in the 2nd quadrant then the standard angle is: θ = 180 + Ⲫ
3. If both the coordinates are negative, then the angle exists in the 3rd quadrant then the standard angle is: θ = 270 + Ⲫ
4. If the x-coordinate is positive, but the y-coordinate is negative, then the standard angle is: θ = 360 +  Ⲫ. 

Let’s under this with the help of examples.

Example 5

Find the direction of a vector directed from origin to the coordinates (6, -7). 

Solution

We will take help from the inverse tangent formula: 

θ = tan-1 (-7/6)

θ = -49.23°

Here we can see from the coordinates of the vector that it was lying in Quadrant IV. 

Now, here is the deal:

The formula gives the shortest angle from either the positive or negative x-axis. The convention is to represent the angle with a positive sign from the positive x-axis. For this, we subtract from 360 ° to the obtained angle. 

θ’ = -49.23 + 360

θ = 310.77°

Example 6

Find the direction of the vector (-4,3).

Solution

By looking at the coordinates, we know that the vector lies in Quadrant II:

θ = tan-1 (3/-4)

θ = -36.87°

This is the angle from the negative x-axis. Now, to get the positive answer, and calculated from the positive x-axis counterclockwise: 

θ = -36.87 + 180

θ = 143.13°

from the positive x-axis in a counter-clockwise direction. 

For Finding The Direction Of The Resultant Vector

Moving on, let’s see how we can find the direction of the resultant of two or more vectors. 

As you know, to calculate the resultant vector of two or more individual vectors, we find their respective rectangular coordinates first. Next, we add the x-component and y-component of the two vectors. The resultant x-component and y component are, in fact, the components of the resultant vector.

Following are the step to calculate the direction of a resultant of two or more vectors: 

Let’s say you have vectors A and B, and you want to find their resultant and direction.

1. Dissolve both vectors into their rectangular components. 
2. We know, R = A + B. Similarly, Rₓ = Aₓ + Bₓ and R𝚢 = A𝚢 + B𝚢
3. Now using the inverse tangent property, replace x and y with x, y-components of the resultant, i.e., =tan-1(Ry/Rx)
4. Determine the quadrant of the resultant and modify theta according to it.

Practice Problems 

1. Find the direction of a vector whose initial and final points are (5, 2) and (4, 3), respectively. 
2. Find the direction of a vector whose initial and final points are (2, 3) and (5, 8), respectively. 
3. A vector is directed from the origin to (7, 4). Find its direction. 
4. Find the direction of a vector whose coordinates are (-7, -5).
5. Find the direction of a vector whose coordinates are (1, -1).

Answers

1. -45° or 135°
2. 59°
3. 29.74°
4. 234°
5. -45° or 135°

All the vector diagrams are constructed by using GeoGebra.

Magnitude and Direction of Vectors

Magnitude of a Vector

Direction of a Vector

The direction of a vector is the measure of the angle it makes with a horizontal line .

One of the following formulas can be used to find the direction of a vector:

How to Find a Vector’s Magnitude and Direction

In physics, when you’re given the vector components, such as (3, 4), you can easily convert to the magnitude/angle way of expressing vectors using trigonometry.

For example, take a look at the vector in the image.

Suppose that you’re given the coordinates of the end of the vector and want to find its magnitude, v, and angle, theta. Because of your knowledge of trigonometry, you know

Where tan thetais the tangent of the angle. This means that

theta = tan^–1(y/x)

Suppose that the coordinates of the vector are (3, 4). You can find the angle theta as the tan^–1(4/3) = 53 degrees.

You can use the Pythagorean theorem to find the hypotenuse — the magnitude, v — of the triangle formed by x, y, and v:

So if you have a vector given by the coordinates (3, 4), its magnitude is 5, and its angle is 53 degrees.

Vectors and Direction

• Vectors and Direction
• Vector Addition
• What is a Resultant?
• Vector Components
• Vector Resolution
• Component Addition
• Relative Velocity and River Boat Problems
• Independence of Perpendicular Components of Motion

A study of motion will involve the introduction of a variety of quantities that are used to describe the physical world. Examples of such quantities include distance, displacement, speed, velocity, acceleration, force, mass, momentum, energy, work, power, etc. All these quantities can by divided into two categories – vectors and scalars. A vector quantity is a quantity that is fully described by both magnitude and direction. On the other hand, a scalar quantity is a quantity that is fully described by its magnitude. The emphasis of this unit is to understand some fundamentals about vectors and to apply the fundamentals in order to understand motion and forces that occur in two dimensions.

Examples of vector quantities that have been previously discussed include displacement, velocity, acceleration, and force. Each of these quantities are unique in that a full description of the quantity demands that both a magnitude and a direction are listed. For example, suppose your teacher tells you “A bag of gold is located outside the classroom. To find it, displace yourself 20

meters.” This statement may provide yourself enough information to pique your interest; yet, there is not enough information included in the statement to find the bag of gold. The displacement required to find the bag of gold has not been fully described. On the other hand, suppose your teacher tells you “A bag of gold is located outside the classroom. To find it, displace yourself from the center of the classroom door 20 meters in a direction 30 degrees to the west of north.” This statement now provides a complete description of the displacement vector – it lists both magnitude (20 meters) and direction (30 degrees to the west of north) relative to a reference or starting position (the center of the classroom door). Vector quantities are not fully described unless both magnitude and direction are listed.

Representing Vectors

Vector quantities are often represented by scaled vector diagrams. Vector diagrams depict a vector by use of an arrow drawn to scale in a specific direction. Vector diagrams were introduced and used in earlier units to depict the forces acting upon an object. Such diagrams are commonly called as free-body diagrams. An example of a scaled vector diagram is shown in the diagram at the right. The vector diagram depicts a displacement vector. Observe that there are several characteristics of this diagram that make it an appropriately drawn vector diagram.

• a scale is clearly listed
• a vector arrow (with arrowhead) is drawn in a specified direction. The vector arrow has a head and a tail.
• the magnitude and direction of the vector is clearly labeled. In this case, the diagram shows the magnitude is 20 m and the direction is (30 degrees West of North).

Conventions for Describing Directions of Vectors

Vectors can be directed due East, due West, due South, and due North. But some vectors are directed northeast (at a 45 degree angle); and some vectors are even directed northeast, yet more north than east. Thus, there is a clear need for some form of a convention for identifying the direction of a vector that is not due East, due West, due South, or due North. There are a variety of conventions for describing the direction of any vector. The two conventions that will be discussed and used in this unit are described below:

1. The direction of a vector is often expressed as an angle of rotation of the vector about its “tail” from east, west, north, or south. For example, a vector can be said to have a direction of 40 degrees North of West (meaning a vector pointing West has been rotated 40 degrees towards the northerly direction) of 65 degrees East of South (meaning a vector pointing South has been rotated 65 degrees towards the easterly direction).
2. The direction of a vector is often expressed as a counterclockwise angle of rotation of the vector about its “tail” from due East. Using this convention, a vector with a direction of 30 degrees is a vector that has been rotated 30 degrees in a counterclockwise direction relative to due east. A vector with a direction of 160 degrees is a vector that has been rotated 160 degrees in a counterclockwise direction relative to due east. A vector with a direction of 270 degrees is a vector that has been rotated 270 degrees in a counterclockwise direction relative to due east. This is one of the most common conventions for the direction of a vector and will be utilized throughout this unit.

Two illustrations of the second convention (discussed above) for identifying the direction of a vector are shown below.

Observe in the first example that the vector is said to have a direction of 40 degrees. You can think of this direction as follows: suppose a vector pointing East had its tail pinned down and then the vector was rotated an angle of 40 degrees in the counterclockwise direction. Observe in the second example that the vector is said to have a direction of 240 degrees. This means that the tail of the vector was pinned down and the vector was rotated an angle of 240 degrees in the counterclockwise direction beginning from due east. A rotation of 240 degrees is equivalent to rotating the vector through two quadrants (180 degrees) and then an additional 60 degrees into the third quadrant.

Representing the Magnitude of a Vector

The magnitude of a vector in a scaled vector diagram is depicted by the length of the arrow. The arrow is drawn a precise length in accordance with a chosen scale. For example, the diagram at the right shows a vector with a magnitude of 20 miles. Since the scale used for constructing the diagram is 1 cm = 5 miles, the vector arrow is drawn with a length of 4 cm. That is, 4 cm x (5 miles/1 cm) = 20 miles.

Using the same scale (1 cm = 5 miles), a displacement vector that is 15 miles will be represented by a vector arrow that is 3 cm in length. Similarly, a 25-mile displacement vector is represented by a 5-cm long vector arrow. And finally, an 18-mile displacement vector is represented by a 3.6-cm long arrow. See the examples shown below.

In conclusion, vectors can be represented by use of a scaled vector diagram. On such a diagram, a vector arrow is drawn to represent the vector. The arrow has an obvious tail and arrowhead. The magnitude of a vector is represented by the length of the arrow. A scale is indicated (such as, 1 cm = 5 miles) and the arrow is drawn the proper length according to the chosen scale. The arrow points in the precise direction. Directions are described by the use of some convention. The most common convention is that the direction of a vector is the counterclockwise angle of rotation which that vector makes with respect to due East.

In the remainder of this lesson, in the entire unit, and in future units, scaled vector diagrams and the above convention for the direction of a vector will be frequently used to describe motion and solve problems concerning motion. For this reason, it is critical that you have a comfortable understanding of the means of representing and describing vector quantities. Some practice problems are available on site at the following web page:

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