You learned in 1-D kinematics that the physical quantities such as displacement, velocity and acceleration are vectors. A vector has a magnitude (size) and a direction. But a physical quantity that does not have a direction is called a scalar. Length, distance, and speed are scalars.
In 1-D motions, we took the object's path as one of the axes ($x$ or $y$), and the object moves either in the positive or in the negative direction on the axis. So, we took the sign as direction for the vectors describing the motion. But if the path is not a straight line as in a 2-D or a 3-D motion, we cannot just use the sign as direction. Further, vectors in 1-D are simple numbers that we can add or subtract like any other numbers. This is not the case in 2 or 3 dimensions. Since we use vectors to describe motions, it is important to learn the properties of vectors in higher dimensions.
In this page, you will learn about the properties of vectors in general with the focus on 2-D. Vectors in 3-D are similar to the vectors in 2-D, but with one more dimension.
Vector notation
A vector is drawn as an arrow. The length of the arrow represents the magnitude of the vector and the arrowhead shows its direction.
Vector A.
When writing a vector, we write it as a letter with an arrow over it or as a thick letter.
i.e., vector A is written as $\vec{A}$ or A.
Magnitude of a vector is the size of the vector, which is a positive number. For vector, $\vec{A}$, magnitude is written with an absolute value symbol, $|\vec A|$ or just $A$, without the arrow.
Direction of a vector
We use the angle, $\theta$, the vector makes with the $+x$ axis as the direction of the vector.
$\theta$ is the direction of $\vec A$.
So, in real world, if you take the east as $+x$ direction, and the angle $\theta$ is 30°, then the direction of the vector, $\vec A$ is 30° north of east.
Equal vectors
For a vector, only the magnitude and the direction are important, position is not. So, if you have two vectors in two different places, but have same magnitude and direction, then they are equal. So, you can move a vector without rotating it or changing its length, still the vector is the same.
Negative vector
Negative of a vector has same magnitude as the original vector, but has exactly opposite direction. In the following figure, vector $\vec{B}$ is on the left and its negative vector is on the right. A vector and its negative vector.
Multiplying a vector with a number
If you multiply a vector with a positive number, you will get another vector of same direction but of different magnitude. For example, if you multiply a positive number, $s$ with a vector, $\vec{A}$, then the magnitude of the new vector will be $sA$. The direction of both the vectors will be the same.
In the figure below, a vector, $\vec C$ and the vector multiplied with a number $2$ are given. The magnitude (length) of the new vector is twice that of the original vector. But both the original and the new vector are in the same direction.
On the other hand, if you multiply a negative number with a vector, then the direction of the new vector will be opposite to that of the original vector. And, the magnitude of the new vector will be the positive of the number times the magnitude of the original vector.
Vector addition
If you add (or subtract) two or more vectors, you will get another vector, called resultant vector. There are two ways to add vectors, the graphical method and the components method. The components method is also called analytical method.
Graphical methods
Assume that a person starts at point K, move to point L, and from there move to another point M as in the figure.
What is the displacement of the person?. The net displacement of the person is a vector drawn from the initial position K to the final position, M.
You see that the person is making two displacements, one from K to L and another from L to M. Let us take the vector from K to L as $\vec A$ and the vector from L to M as $\vec B$. And the net displacement vector as $\vec R$. Now, if you add the two displacement vectors, you will get the net displacement vector,
$\vec R = \vec A+\vec B$
We call the vector, $\vec R$ the resultant vector of $\vec A$ and $\vec B$ as it is the sum of the two vectors.
In the figure, you see that the tail (the end with no arrowhead) of the second vector, $\vec B$ is connected to the tip (the end with the arrowhead) of the first vector, $\vec A$ and the resultant vector, $\vec R$ is the vector drawn from the tail of the first vector to the tip of the second vector. Based on these, now we can create a method of adding vectors, which is called tail-to-tip method.
Tail-to-tip method
We have two vectors, $\vec A$ and $\vec B$ and want to add them.
To add, keep the first vector, $\vec A$ and draw (or move) the second vector, $\vec B$ so that its tail is connected to the tip of the first vector.
Now draw a vector from the tail of the first vector to the tip of the second vector, which is the resultant vector, $\vec R$.
This graphical method is called tail-to-tip method of vector addition.
In vector addition, order is not important. You can switch $\vec A$ and $\vec B$, i.e., take $\vec B$ as the first vector and $\vec A$ as the second vector, then apply the tip-to-tail method, still you get the same resultant vector:
i.e., $\vec A+\vec B=\vec B+\vec A$.
This is called commutative property of vector addition.
The tail-to-tip method can be used to add any number of vectors. If you have more than two vectors, connect the tail of the second vector to the tip of the first vector, and connect the tail of the third vector to the tip of the second vector and so on. Then draw a vector from the tail of the first vector to the tip of the last vector to get the resultant vector.
Parallelogram method
You can also get the resultant vector, if you do the following. Move the vector, $\vec B$ and connect its tail to the tail of the vector, $\vec A$. That is both the vectors start from a common point. Next, take the two vectors as the adjacent sides, complete a parallelogram. Now draw a vector from the common point along the diagonal of the parallelogram . This vector is the resultant vector, $\vec R$.
This way of adding two vectors is called the parallelogram method. The parallelogram method works if there are only two vectors to add. If there are more than two vectors, then you cannot use the parallelogram method.
Addition of parallel vectors
If two different vectors point to the same direction or exactly in opposite directions, they are called parallel vectors.
Addition of parallel vectors of same direction
We have two parallel vectors $\vec A$ and $\vec B$ of same direction as shown in the figure below. To add them by the tail-to-tip method, move the vector $\vec B$ and connect its tail with the tip of the vector, $\vec A$. Now, the resultant vector is the vector drawn from the tail of $\vec A$ to the tip of $\vec B$.
You can see that direction of the the resultant, $\vec R$ is same as that of the original vectors. And the magnitude of the resultant is the sum of the magnitudes of the two vectors,
magnitude of $\vec R = $ magnitude of $\vec A+$ magnitude of $\vec B$.
i.e., $R=A+B$.
Addition of parallel vectors of opposite directions
The figure below shows the addition of two vectors of opposite directions. The vectors $\vec A$ and $\vec B$ are parallel vectors of opposite directions. If you add them by the tail-to-tip method, the resultant vector is the vector drawn from the tail of $\vec A$ to the tip of vector $\vec B.$
The direction of the resultant is the direction of the vector, $\vec A$, which is the larger of the two vectors and the magnitude of the resultant is $A-B$, which is the difference of the magnitudes of the two vectors.
i.e., $R=|A-B|$.
An absolute value symbol is added to make sure, the magnitude, R is a positive number.
Components method
The tail-to-tip method shows you the picture of adding two or more vectors. If you draw the vectors with the correct scale with a ruler and protractor, then with this method you can measure the magnitude and the direction of the resultant vector. But with the components method, we can find the magnitude and the direction of the resultant vector without any drawing tools.
First, you will learn about vector components and then you will learn how we can use the components to add vectors.
Components of a vector
Let us consider a vector, $\vec A$, which makes an angle $\theta$ with the $+x$-axis.
Now, draw a rectangle with the vector as one of the diagonal as shown.
Next, draw two vectors, one along the $x$ axis on the length of the rectangle and the other along the $y$ axis on the height of the rectangle. Let us call the vector on the $x$ axis, $\vec A_x$ and the vector on the $y$ axis, $\vec A_y$.
Now, you can realize that the vector $\vec A$ is just the sum of the vectors, $\vec A_x$ and $\vec A_y$ according to the parallelogram method,
$\vec A=\vec A_x+\vec A_y$
Thus, any vector, $\vec A$ can be written as a sum of two perpendicular vectors, one along the $x$ axis and the other along the $y$ axis. We call the vectors, $\vec A_x$ and $\vec A_y$, the components or component vectors of $\vec A$. The component vectors are 1-D vectors as they are restricted to only one of the axes. So, we ignore the vector sign for component vectors. And, we take $A_x$ as the $x$ component of vector, $\vec A$ and $A_y$ its $y$ component.
Finding the components of a vector
You learned that we can move a vector without rotating or changing its length. In the figure below, you can see that I moved the $y$ component vector from the left side of the rectangle to the right hand side.
Now, we have a right angled triangle, OPQ, with $A$ as the hypotenuse. And the sides are the the vector components, $A_x$ and $A_y$. If you take the cosine and the sine of the angle $\theta$, you will get,
$\cos\theta =\dfrac{A_x}{A}$ and $\sin\theta =\dfrac{A_y}{A}$
Solving for $A_x$ and $A_y$,
$\boxed{A_x=A\,\cos\theta\:}$ and
$\boxed{A_y=A\,\sin\theta}$
With these equations, you can find the components of a vector $\vec A$ if you know its magnitude, $A$ and its direction, $\theta$.
Now, applying the Pythagorean theorem to the right triangle OPQ, you will get,
$\boxed{A=\sqrt{A_x^2+A_y^2}}$
This equation tells you that you can find the magnitude of a vector, if you know its components.
Now, take the tangent of the angle $\theta$, you get
$\boxed{\tan\theta=\dfrac{A_y}{A_x}}$
With this equation, you can find the direction, $\theta$ of the vector $\vec A$ from its components. To find the $\theta$, first find the tangent using the equation, then take the inverse tangent.
It is important to note that, the vector, $\vec A$ is in the first quadrant. A vector can be in any one of the four quadrants. In the first quadrant, both the vector components are positive as the $x$ component point to the right, the positive $x$ direction and the $y$ component point upward, the positive $y$ direction. But if the vector is in another quadrant, this is not the case. For example, in the figure below, I have a vector, $\vec P$ that is in the second quadrant. By drawing the components of the vector, you can see that the $x$ component of $\vec P$ is in the negative $x$ direction, and the $y$ component is in the positive $y$ direction. So, $P_x$ is negative, and $P_y$ is positive.
So, you must identify the sign of the components from the figure and add appropriate sign to the equation when finding the components. If a component points to the negative direction, then you must add a minus sign. So, for the vector $\vec P$, we have
$P_x=-P\,\cos\theta$ and $P_y=P\,\sin\theta$
Components of a vector that is on one of the axes
A vector can also be on one of the axes (or parallel to an axis). The vector, $\vec F$ in the figure below is on the $x$ axis.
So, the angle the vector makes with the $+x$ axis, $\theta$ is zero. Thus, the vector has no $y$ component. Since the vector points to the positive $x$ direction, its $x$ component is positive. So, the components of the vector $\vec F$ are
$F_x=F$ and $F_y=0$
In the figure below, the vector, $\vec D$ is on the $y$ axis. So, this vector has no $x$ component. Since the vector points to the negative $y$ direction, the $y$ component of the vector is negative.
So, the components of the vector $\vec D$ are
$D_x=0$ and $D_y=-D$
Adding vector with components
In the figure, you see that there are two vectors, $\vec A$ and $\vec B$, you need to add them and find the resultant vector. Finding a vector means, finding the magnitude and the direction of the vector. Using the tip-to-tail method, you can draw the resultant vector, $\vec R$ as in the figure. Vector addition
Now, the question is, what is the magnitude and the direction of the resultant vector?. We can find them by using the components of the vectors, $\vec A$ and $\vec B$ as you will see.
First, let us draw the $x$ and the $y$ components of the vectors, $\vec A$ and $\vec B$.
Next, add the $x$ components of the two vectors by the tail-to-tip method. Likewise, add the $y$ components of the vectors. You will get the following figure.
Now, you can see in the figure that the $x$ component of the resultant vector is just the sum of the $x$ components of the two vectors. That is
$R_x=A_x+B_x$
And the $y$ component of the resultant is the sum of the $y$ components of the two vectors:
$R_y=A_y+B_y$
Once you get the $x$ and the $y$ components of $\vec R$, you can find its magnitude and direction by the equations:
$R=\sqrt{R_x^2+R_y^2}$
and
$\tan\theta=\dfrac{R_y}{R_x}$
We have added two vectors by the components method. But you can add any number of vectors with the components method. For example, if you have 3 vectors, $\vec A$, $\vec B$ and $\vec C$. Then the resultant (sum) of the three vectors is.
$\vec R=\vec A+\vec B+\vec C$,
To get the $x$ component of $\vec R$, find the $x$ components of all the three vectors and add them. That is
$R_x=A_x+B_x+C_x$;
And, to get the $y$ component of $\vec R$, find the $y$ components of all the three vectors and add them. That is
$R_y=A_y+B_y+C_y$;
Now you may have a question, what I need to do if I am subtracting one of the vectors, say $\vec B$ instead of adding. That is
$\vec R=\vec A-\vec B+\vec C$.
In this case you just need to take the opposite (change the sign) of the $x$ and the $y$ components of $\vec B$, when adding to find the components of the resultant. So,
$R_x=A_x-B_x+C_x$;
$R_y=A_y-B_y+C_y$;
Problem: A truck travels 36 km in a direction 32° east of north. Next, it travels 52 km due south. Finally, it travels 64 km in a direction 28° north of west. (a) Determine how far the truck ends up from the starting point, and (b) find the direction of the truck's displacement.
Solution:
There are three displacements for the truck. Each displacement is a vector. Let us label the vectors as $\vec A$, $\vec B$ and $\vec C$. If you draw these vectors, you will get the following figures,
Assuming the truck started with the origin, and adding all the vectors using tail-to-tip method, you get the resultant vector, $\vec R$.
The net displacement of the truck is the resultant vector, $\vec R$ drawn from the starting position to the final position of the truck. We need to find how far the truck is from the starting point, which is the magnitude, $R$. And the direction of the truck's net displacement, which is $\theta$ of the resultant vector.
First step is, you need to find the $x$ and the $y$ components of all the three vectors $\vec A$, $\vec B$ and $\vec C$.
Take the vectors one by one, draw and find their $x$ and $y$ components.
$\underline{\vec A}$:
Magnitude of the vector is $A= 36 \,km$.
The components, $A_x$ and $A_y$ both are positive as $A_x$ is in the $+x$ direction and $A_y$ is in the $+y$ direction. The angle of the vector, $\vec A$ is given from the $y$ axis. But in the equations, we need to use the angle of the vector from the $x$ axis. To find the angle from the $x$ axis, you need to subtract the angle from 90°. So, the angle of the vector from the $x$ axis is 90°- 32° = 58°.
Magnitude of the vector is $B= 52 \,km$. The vector is on the $y$ axis, so its $x$ component is zero. Further, the vector is on the negative $y$ axis, so the $y$ component is negative:
$B_x=0$
$B_y=-B=-52\,km$
$\underline{\vec C}$:
Magnitude, $C= 64 \,km$; $\theta = 28^\circ$ from the $x$ axis.
$C_x$ is negative as it is on the negative $x$ axis, $C_y$ is positive as it is on the positive $y$ axis. Therefore,
$C_x=-C \cos\theta= -64\,km \cos 28=-56.51 \,km$
$C_y=+C \sin\theta= 64\,km \sin 28=30.05 \,km$
Now, add all the $x$ components of the vectors, you will get the $x$ component of the resultant, $R_x$. And, add all the $y$ components, you will get the $y$ component, $R_y$.
$R_x=A_x+B_x+C_x$
$=19.08 km+0-56.51km$
$=-37.43km$
$R_y=A_y+B_y+C_y$
$=30.53 \,km-52 \,km+30.05 \,km$
$=8.58 \,km$
Now, we can find the magnitude and the direction of the resultant.
Magnitude of the resultant is
$R=\sqrt{R_x^2+R_y^2}$
$=\sqrt{(-37.43)^2+(8.58)^2} \:km$
$=38.4 \,km$
Rounding to the correct number of significant figures,
$\boxed{R=38\, km}$
Now, find $\theta$ by using the equation:
$\tan \theta = \dfrac{R_y}{R_x}$
$=\dfrac{8.58}{-37.43}$
$=-0.229$
or
$\boxed{\theta = -13^\circ}$
$\theta$ is negative means, the vector, $\vec R$ can be below the positive $x$ axis or above the negative $x$ axis. To find which one, we need to draw the components of $\vec R$. Then, from the components draw the $\vec R$ by forming a rectangle.
We get, $\vec R$ is above the negative $x$ axis. The tail-to-tip method also confirms this.
Since the $-x$ direction is the west, the vector, $\vec R$ is 13° from the west, towards the north. So, the direction of the truck's displacement is 13° north of west.
Thus, the answers are:
(a) the truck ends up at 38 km from the starting point.
(b) the direction of the net displacement of the truck is 13° north of west.
