Physics: Scalars and Vectors
Physical quantities are classified as either scalars or vectors. Basically, the difference lies in the direction that is associated with a vector but not with a scalar. A scalar quantity is a quantity with magnitude only. It is specified completely by a single number, along with the proper unit. For example, the distance between two points, the mass of an object, the temperature of a body and the time at which a certain event happened, are all scalars. The rules for combining scalars are the rules of ordinary algebra. Scalars can be added, subtracted, multiplied and divided just as the ordinary numbers, although addition and subtraction make sense only with quantities with same units.
Examples of Scalar Quantities (scalars)
- If the length and breadth of a rectangle are 1.0 m and 1.5 m respectively, then its perimeter is the sum of the lengths of the four sides, 1.0 m + 1.5 m +1.0 m + 1.5 m = 5.0 m. The length of each side is a scalar and the perimeter is also a scalar.
- The maximum and minimum temperatures on a particular day are 35.6 °C and 24.2 °C respectively. Then, the difference between the two temperatures is 11.4 °C.
- A uniform solid cube of aluminum of side 10 cm has a mass of 2.7 kg, then its volume is 10-3 m3 (a scalar) and its density is 2.7 x 103 kg m-3 (a scalar).
A vector quantity is a quantity that has both a magnitude and a direction and obeys the triangle law of addition or equivalently the parallelogram law of addition (as discussed in the next section). Some physical quantities that are represented by vectors are displacement, velocity, acceleration and force.
Examples of Vector Quantities
- Suppose an ant walks 3 m to the north, turns right, and walks 4 m in the east direction. Then the total path length (scalar) of the ant is 7 m, while the displacement (vector) of the ant from its initial position is 5 m in a direction which is about 36.87 degrees north of the east.
- Raindrops in a certain area fall vertically down at 3 m/s on the ground when there is no wind. But you can imagine that they won’t fall vertically when there is a wind blowing, e.g. a horizontal wind with a velocity 3 m/s would cause the raindrops to fall at m/s, 450 from the vertical.
Note on vector representation: To represent a vector, we use a bold face type here. Thus, a velocity vector can be represented by a symbol v. (When written by hand, a vector is often represented by an arrow placed over a letter). The magnitude of a vector is often called its absolute value, indicated by |v| = v. Thus, a vector is represented by a bold face, e.g. by A, a, r, x, y, with respective magnitudes denoted by light face A, a, r, x, y.
Multiplying a vector A with a positive number λ gives a vector whose magnitude is changed by the factor λ but the direction is the same as that of A:
|λ A| = λ |A| if λ > 0.
For example, if A is multiplied by 2, the resultant vector 2A is in the same direction as A and has a magnitude twice of |A| as shown in Fig. 1.1.
Multiplying a vector A by a negative number λ gives a vector λA whose direction is opposite to the direction of A and whose magnitude is –λ times |A|.
|λ A| = -λ |A| if λ < 0.
This procedure of vector addition is called the triangle method of vector addition, as the two vectors and their resultant form three sides of a triangle. If we find the resultant of B + A as in Fig. 2.2(b) the same vector R is obtained. Thus, vector addition is commutative:
A + B = B + A (2.1)
The addition of vectors also obeys the associative law as illustrated in Fig. 2.3. The result of adding vectors A and B first and then adding vector C is the same as the result of adding B and C first and then adding vector A:
(A + B) + C = A + (B + C) (2.2)
What is the result of adding two equal and opposite vectors? Consider two vectors A and –A. Their sum is A + (–A). Since the magnitudes of the two vectors are the same, but the directions are opposite, the resultant vector has zero magnitude and is represented by 0 called a null vector or a zero vector:
A – A = 0 |0|= 0 (2.3)
Since the magnitude of a null vector is zero, its direction cannot be specified. The null vector also results when we multiply a vector A by the number zero. The main properties of 0 are:
A + 0 = A; λ 0 = 0; 0 A = 0.
Subtraction of vectors can be defined in terms of addition of vectors. We define the difference of two vectors A and B as the sum of two vectors A and –B:
A – B = A + (–B) (2.4)
It is shown in Fig 2.4. The vector –B is added to vector A to get R2 = (A – B). The vector R1 = A + B is also shown in the same figure for comparison.
We can also use the parallelogram method to find the sum or difference of two vectors. Suppose we have two vectors A and B. To add these vectors, we bring their tails to a common origin O as shown in Fig. 2.5(a). Then we draw a line from the head of A parallel to B and another line from the head of B parallel to A to complete a parallelogram OQSP. Now we join the point of the intersection of these two lines to the origin O. The resultant vector is directed from the common origin O along the diagonal (OS) of the parallelogram [Fig. 2.5(b)].
In Fig.2.5(c), the triangle law is used to obtain the resultant of A and B and we see that the two methods yield the same result. Thus, the two methods are equivalent.
Notice that if A and B are equal in length, the parallelogram is a rhombus, and therefore A+B is perpendicular to A-B. Similarly, if A is perpendicular to B, then A+B is equal in length to A-B.
Adding and Subtracting Vectors
As mentioned previously, vectors, by definition, obey the triangle law or equivalently, the parallelogram law of addition. Let us consider two vectors A and B that lie in a plane as shown here:
The lengths of the line segments representing these vectors are proportional to the magnitude of the vectors. To find the sum A + B, we place vector B so that its tail is at the head of the vector A, as in Fig. 2.2(a). Then, we join the tail of A to the head of B. This line OQ represents a vector R, that is, the sum of the vectors A and B.