Physics

Introduction to Vectors for everyone



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We classify quantities into two major groups in Physics. They are vector quantities and scalar quantities. What is the difference? The first ones have both magnitude and direction. The later have only magnitude. Magnitude is the number mentioned for a measure.

The examples for a vector quantity are displacement, velocity, force, momentum and torque. The examples for a scalar quantity are distance, speed, power, work and temperature. Well, one may be confused between displacement and distance. Once more, what distinguishes velocity from speed?

Displacement is a vector quantity but distance is a scalar one. I am glad to give you a simple illustration. Imagine you walk around in a park. You start from a chair in some part of the circle. After a few minute you back to your chair to rest. Physicist will say that your displacement is zero. Even though you have travelled several meters, because you stop at the same point of your departure, your displacement is zero. The distance you have travelled is not zero. The distance is the length of path you have travelled.

Velocity is a vector because it has direction and magnitude. If we say a velocity of an object, we have to tell its magnitude and direction. For example, the plane moves 200 km per hour 600 from North to the West. It is velocity. When someone says the plane moves 200 km per hour. It is speed, for there is no direction mentioned.

Vector denoted by a bold character or a character with a small arrow upper it. We can use a small line in spite of a small arrow. The magnitude of a vector denoted by a normal letter (not in bold), or a letter in an absolute sign, like |A|.

Can we add two or more vectors ? Yes, we can add two or more vectors. The sum of vectors also called the resultant. You have to be careful, because the addition rule is not as one in scalar. 3 m of ribbon plus 4 m of ribbon will give us 7 meters of ribbon. But 3 N and 4 N forces directed perpendicular each other will not give us 7 N force. What's wrong ? The sum of two vectors not only depends on the magnitude of each vector. It also depends on the cosine of the angles between them.

To obtain the sum of two vectors, you can follow this procedure. Add the square of the magnitude of the two vectors. Simply denote it as A1. Multiply the magnitude of the two vectors, and multiply again with the 2 times of the cosine of the angle between the two vectors. Denote the result of the multiplication as A2. Add A1 and A2 denote as A3. The sum of the two vectors is the square root of A3.

The direction of the resultant vector can be determined using sine rule. The direction is referred to the on of the vectors added.

Running from zero to 180 degrees of angle, the sum of two vectors is decreasing. The sum reaches maximum when the two vectors in the same direction. The minimum is reached when the two vectors are in opposite direction. I will back to our example before. The two vectors are perpendicular each other. Then the angle between them is 90 degrees. The sum is equal to the square root of the sum of each square. A cosine 90 degree is equal to zero. The angle seems to be not important here.

In two dimensions, a vector is expressed as the sum two vector components. One is the horizontal component. The other is the vertical component. The horizontal direction is represented by the unit vector i. The unit vector j represented the vertical one. For example the force vector which comprises of 4 unit horizontal and 3 unit vector to the vertical is written as 4i + 3j.

Expressing a vector into components in a certain direction is called resolving a vector. Resolving vectors into their vertical and horizontal components is an easy way to add 3 vectors or more. Simply add all the horizontal components and vertical components respectively to get two single vectors perpendicular each other. We can add these two vectors as our example before to obtain the sum of all vectors.

The direction of the resultant vector is the inverse tangent of the quotient of vertical component to the horizontal component.

Well, we have already known the vector addition. What about the subtraction? A vector can be subtracted to another vector. Suppose we want to subtract vector A from vector B. Simply reverse the direction of vector A. Add this reversed vector with vector B, to get the substation of B - A.

We can multiply two vectors too. There are two kinds of multiplication with vectors, the dot product and the cross product. The dot product gives us a scalar result. The cross product will result a vector. Work is resulted from the dot product. It is the dot product between the force and the displacement. Torque is resulted from the cross product. Torque about a pivot is the product between the force applied and a line vector. The line vector here is running from the pivot to the point where the force applied.

The dot product of two vectors is equal to the product of each magnitude and the cosine of the angle between the vectors. The magnitude of the vector resulted from cross product is equal the product of each vector and the sine of the angle between the two vectors.

A vector can be multiplied algebraically with a number. This will result a vector. We can double a velocity of an object. This means we multiply velocity vector with a scalar 2.

Vector is an important thing in physics. Many difficult calculations can be simply solved using vector analysis. Many concepts involve vectors, so understanding vector is a key to conquer physics.

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