Vectors in physics. Properties of vectors. Addition, subtraction, division, multiplication operations by vectors. The subject expression and solved questions.

Definiton Of Vector

Quantities having both direction and magnitude are called vectors.

Quantities are divide into two categories.

A) Scalar Quantities.

B) Vectorial Quantities

A) Scalar Quantities

These quantities can be described with a magnitude and an unit.

For example; 5 kg of water, 20 kg of bananas, 10 liter of oil, 1 ton of sand.

B) Vectorial Quantities

These quantities are decribed with both magnitude and direction. The magnitude has an unit.

For example; 5 N of force that applied to in the East direction. 2 m/s of acceleration in the West direction, 100 km/h of velocity in the North direction.

The vectors are usually represented by an arrow. A vector have a tail and a head.


Properties Of Vectors

1- A vector can be carried to any point without changing its magnitude and direction.

2- The vectors that its direction and magnitude are equal are equal.

3- The magnitude of vectors is always positive. The negative sign of a vector means that the vector is opposite direction relative to directin of the reference vector.

4- To multiply a vector by scalar number, with magnitude of the vector is multiplied by the scalar number.

5- Vectors whose magnitudes are equal and directions are opposite are called opposite vectors.

6- For to do the subtraction operation between two vectors, the vector that be subtracted from the other vector is inverted and two vectors are adding.

Vector Addition

1- The Vectors That Have Same Direction.

Magnitudes of these vectors are added, its directions are not changed.



As seen in the figure above, the F1 force is 30 N and the F2 force is 50 N.

Find resultant (sum) of these vectors.


To add the vectors that have the same directions, magnitudes of these vectors are added.


F12 = F1 + F2

F12 = 30 + 50

F12 = 80 N

The magnitude of the sum vector is 80 N and its direction is the same with directions of the F1 and F2.

2- Subtracted Of The Vectors That its have the Same Direction

For to subtract the vectors, the vector that be subtract is inverted and done the addition operation.

When a vector is inverted, the direction of the vector is be opposite and its sign is be negative.



The F1 is 26 N and the F2 is 45 N.

Find the F1 – F2 vector.


We are inverse the F2 vector and subtract the magnitudes.


The direction of the resultant vector is the direction of the vector that its scalar value is biggest.

F12 = 26 – 45

= -19 N

The magnitude of the resultant vector is 19 N and its direction is the direction of the F2.

3- Head To Tail Method

In this method, head of a vector is added to the tail of the other vector. The arrow that drawn from tail of the first vector to the head of the last vector is the resultant vector.

In this method, usually, the vectors are drawn on paper divided into unit squares.



In the above figure, the distance of each square is equal 10 N.

What is resultant of the K, L, M and N vectors.


Let we use the head to tail method.

We starting with any vector can add the vectors.


Addition of the K, L, M and N vectors is seen the above figure.

The vector R is the resultant vector. The vector R is five square length.

R = 5•10

R = 50 N

4- Addition Of Vectors Using Cosine Theorem

If the angle between two vectors is α, the resultant vector can found as the follow.

Vt2 = V12 + V22 + 2•V1•V2•cosα



It is seen F1 and F2 vectors in the above figure.

Magnitude of the F1 is 80 N, magnitude of the F2 is 60 N.

The angle between F1 and F2 is 60 degrees.

Find the magnitude of the resultant vector.

 (cos60 = 0,5)


F2 = F12 + F22 + 2F1F2cos60°

F2 = 6400 + 3600 + 2•80•60•0.5

F2 = 14800

F = 121.7 N



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