What is Vector Addition?
The geometrical sum of two or more vectors is known as vector addition. Since the vectors do not follow regular laws of algebra, special formulas are used to perform vector addition. The Triangle law of vector addition is one of the vector addition laws. The composition of a vector is known as the resultant vector.
For any vector addition, there are a few conditions that apply, and that are:
You can’t add vectors and scalars together.
The same nature of scalars should be added; for example, we can add distance with distance, not with time.
The same nature of vectors should be added; for example, we can add velocity with velocity, not with force.
There are basically two laws of vector addition, they are:
Vector addition by parallelogram law
Vector addition by triangle law
Read More: Scalar and Vectors
What is the triangle law of vector addition?
A mathematical concept that is used to find the sum of two vectors is the triangle law of vector addition. When the head of the first vector is combined with the extremity of the second vector and then linking the head of the second vector to the tail of the first vector to form a triangle, we can find the resultant sum vector.
Important Questions on Triangle Law of Vector Addition
1) What will be the direction and magnitude of the resulting vector, if P and Q are two vectors making an angle of 45° with each other, and have scales (magnitudes) of 5 and 10 units, respectively.
Using the triangle law,
To find the direction of the resultant vector, Ο = tan-1[(Q sin θ)/ (P + Q cos θ)]
= tan-1[(10 sin 45°)/ (5 + 10 cos 45°)]
= tan-1[(7.07)/ (12.07)]
= 30.35°
Now,
|R| = √ (P2 + Q2 + 2PQ cos θ)
= √ (52 + 102 + 2 × 5 × 10 cos 45°)
= √ (125 + 50√2)
= 13.989 units
2) Find the angle between the two vectors A and B acting on a body, if the resulting vector has a value of √15 units and their magnitudes are 3 and √3 units.
Let the two given vectors to be |A| = 2 and |B| = √2
According to the formula:
R = √ (A2 + 2AB cos θ + B2)
⇒ √15 = √ (32 + 2 × 3 × √3 cos θ + (√3)2)
⇒ 15 = 9 + 6√3 cos θ + 3
⇒ 6√3 cos θ = 15 – 12
⇒ θ = cos-1 [3/ (6√3)]
⇒ θ = cos-1(0.288)
⇒ θ = 73.22°
3)What is the triangle law of vector addition?
A mathematical concept that is used to find the sum of two vectors is the triangle law of vector addition. When the head of the first vector is combined with the extremity of the second vector and then linking the head of the second vector to the tail of the first vector to form a triangle, we can find the resultant sum vector.
4) When two vectors in the same direction are added, what will be the magnitude of the resulting vector?
The lengths of the vectors will be added when two vectors in the same direction are added to
each other. Hence, the resultant vector will accept the resultant length where
the magnitude of the vector is length. Therefore, to give the magnitude of the resultant vector, the magnitudes add on.
5) What is the difference between triangle law and parallelogram law of vector addition?
In parallelogram law, the resultant sum vector is given by the diagonal of a parallelogram, and in the triangle law of vector addition, it is given by the third side of the triangle. Both the laws give a similar magnitude and direction of the resultant sum vector; therefore, triangle law and parallelogram law of vector addition are equivalent methods of vector addition.
6) Calculate the magnitude of the resultant vector; if two vectors P and Q of magnitude 6 units and 8 units respectively, make an angle of 60o.
By following the triangle law of vector addition, the resultant vector is given by:
R = P + Q
The magnitude of R is:
R=|R|=√82+62+2*8*6cos60o
R=√64+36+48
R=√148units
7) Find the addition of vectors XY and YZ, where XY = (5, 7) and YZ = (3, 9).
We will perform the vector addition by adding their corresponding components of the given vectors:
XY + YZ = (5, 7) + (3, 9)
= (5 + 3, 7 + 9) = (8, 16).
8) What is the polygon law of vector Addition?
Polygon law of vector addition says that by the side of a polygon taken in the same order, if the number of vectors can be signified in magnitude as well as direction, then by the closing side of the polygon taken in the opposite order, their results can be represented in magnitude along with the direction.
9) What are the properties of vector addition?
There are two important properties of vector addition:
1. Commutative law – There is no particular order of addition in vectors; for example, P+Q is equal to Q+P.
2. Associative law – This law states that the sum of three vectors is independent of which pair of vectors is added first, i.e. (P+Q) +R = P+(Q+R).
10) What is the vector addition rule?
Vector addition rules are the rules to be followed while adding vectors.
The condition rules are –
1. Only the same nature of vectors can be added. For example, acceleration must be added with acceleration only and not mass.
2. Vectors and scalars together can’t be added.
Practice Questions
1) What is the triangle law of vector addition?
2) What is the parallelogram law of vector addition?
3) What is the difference between the triangle and parallelogram law of vector addition?
4) What is the polygon law of vector addition?
5) What are the rules for addition in vectors?
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