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Properties of vector addition are related to the addition of vector quantities. You've probably heard of scalar and vector quantities in Physics terminology. A magnitude is frequently used to define any physical quantity. As a result, a vector is defined as a quantity with magnitude and direction. Vector quantities include force, linear momentum, velocity, weight, and so on.
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Key Terms: Force, linear momentum, velocity, weight, two-dimensional vectors, vector's direction, parallel translation.
Common Depictions of Vectors
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An arrow drawn over a signifying symbol denotes a vector. For instance, \(\vec {a}\), or \(\vec {b}\). The magnitude of the vectors \(\vec {a}\) and \(\vec {b}\) is represented by the letters ||a|| and ||b||, respectively. Because force represents the magnitude of intensity or strength exerted in one direction, it is a vector. Velocity is a vector whose speed represents the magnitude with which an object moves along a specific path.
Common depiction of vector
Depiction of Two-Dimensional Vectors
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Two-dimensional vectors can be expressed in two ways: Geometric form, Rectangular notation, and Polar notation.
Vector Depiction in Geometry
In simple terms, a line with an arrow represents a vector, with the length of the line representing the magnitude of the vector and the arrow pointing in the vector's direction.
Depiction of Vector
Rectangular Depiction
This vector's rectangular coordinate notation is \(\vec {v}\) = (6,3)
The usage of two-unit vectors î = (1,0) and = (0,1) in the form v = 6î + 3\(\hat {j}\) is an alternative notation.
Rectangular Depiction
Polar Depiction
We define the vector magnitude r, r0 and angle with the positive x-axis in polar notation.
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Equality of Vectors
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When two vectors with the same magnitude and direction are compared, they are identical vectors. As a result, a vector does not change its original vector if it is translated to location without changing its direction or rotating, i.e., parallel translation. The vectors before and after the position change are equal. Nonetheless, it's better if you recall that vectors of the same physical quantity should be compared. For example, the Force vector of 10 N in the positive x-axis and the velocity vector of 10 m/s in the positive x-axis might be equated.
Equality of Vectors
Properties of Vector Addition
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The vector addition is defined as the sum of two or more vectors. Triangle law and parallelogram law are two significant laws related to vector addition. Similarly, the properties of vector addition are as follows:
- Commutative Property
- Associative Property
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Commutative Property of Vector Addition
“For any two vectors, commutative property of vector addition asserts that for any vectors \(\vec {a}\) and \(\vec {b}\),
\(\vec {a} + \vec {b} = \vec {b} + \vec {a}\)
Community property
Let \(\overline {|AB|}\) = \(\vec {a}\) and \(\overline {|BC|}\) = \(\vec {b}\)
We can now write, utilizing the triangle law of vector addition from the triangle ABC:
|AC|= \(\vec {a}\) + \(\vec {b}\)
We have a parallelogram with opposite sides that are parallel and equal,
|AD| = |BC| = \(\vec {b}\)
Using the triangle law from the triangle ADC once more, we get
|AC| = |AD| + |DC|= \(\vec {b}\) + \(\vec {a}\)
Therefore,
\(\vec {a} + \vec {b} = \vec {b} + \vec {a}\)
The commutative property of vector addition is thus proved.
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Associative Property of Vector Addition
The adding of three vectors is irrespective of the pair of vectors added first in this property.
(a + b) + c = a + (b + c)
“For any three vectors \(\vec {a}\), \(\vec {b}\) and \(\vec {c}\), the associative property of vector addition asserts that,
\((\vec {a} + \vec {b}) + \vec {c} = \vec{a} + (\vec{b} + \vec {c} )\)
Assume that the vectors \(\vec {a}\), \(\vec {b}\) and \(\vec {c}\) are represented by PQ, QR and RS.
Associative Property
Thus,
\(\vec {a} + \vec {b}\) = PQ + QR = PR
Also,
\(\vec {b} + \vec {c}\) = QR + RS = QS
Hence,
\(\vec {a} + \vec {b}\) + \(\vec {c}\) = PR + RS = PS, and
\(\vec {a}\) + \(( \vec {b}+ \vec {c} )\) = PQ + QS = PS
That is,
\(\vec {a} + \vec {b}\) + \(\vec {c}\) = \(\vec {a}\) + \(( \vec {b}+ \vec {c} )\)
Hence, the associative property of vector addition is proved in this case.
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The video below explains this:
Addition of Vectors Detailed explanation:
Things to Remember
- A vector quantity with both magnitude and direction. As a result, a vector is defined as a quantity with magnitude and direction.
- Vectors can be expressed in two dimensions in two ways: geometric form, rectangular notation, and polar notation.
- When two vectors with the same magnitude and direction are compared, they are identical vectors. As a result, a vector does not change its original vector if it is translated to location without changing its direction or rotating, i.e., parallel translation.
- According to Commutative Property of Vector Addition, “For any two vectors, commutative property of vector addition asserts that for any vectors \(\vec {a}\) and \(\vec {b}\), \(\vec {a}\) + \(\vec {b}\) = \(\vec {b}\) + \(\vec {a}\).
- According to Associative Property of Vector Addition, “For any three vectors a, b and c, the associative property of vector addition asserts that
(\(\vec {a}\) + \(\vec {b}\)) + \(\vec {c}\) = \(\vec {a}\) + ( \(\vec {b}\) + \(\vec {c}\) ).
Sample Questions
Ques. Consider\(\vec {a}\) = \(\hat {i}\) + 2\(\hat {j}\) and b = 2\(\hat {i}\) + \(\hat {j}\). Is \(\mid{\vec{a}\mid}\) = \(\mid{\vec{b}\mid}\) ? Are the vectors \(\vec {a}\) and \(\vec {b}\) equal? (2 marks)
Ans. We know that \(\mid{\vec{a}\mid}\) = \(\sqrt{1^2+2^2}\) = \(\sqrt{5}\) and \(\mid{\vec{b}\mid}\) = \(\sqrt{2^2+1^2}\)=\(\sqrt{5}\)
So, \(\mid{\vec{a}\mid}\) = \(\mid{\vec{b}\mid}\)
Since the corresponding components of both vectors are distinct, the two vectors are not equal.
Ques. There is a unit vector in the direction of \(\vec {a}\) = 2\(\hat {i}\) +3\(\hat {j}\)+ \(\hat {k}\). Find it. (2 marks)
Ans. The unit vector in the direction of vector \(\vec {a}\) is defined as \(\hat {a}\)= \(\frac {1} {\mid{\vec{a}\mid}}\)\(\vec {a}\)
We know that \(\mid{\vec{a}\mid}\) = \(\sqrt{2^2+3^2+1^2}\) = \(\sqrt{14}\)
Therefore, \(\hat {a}\)= \(\frac {1}{\sqrt{14}}\) (2\(\hat {i}\)+3\(\hat {j}\)+ \(\hat {k}\))
= \(\frac {2}{\sqrt{14}}\)\(\hat {i}\) + \(\frac {3}{\sqrt{14}}\)\(\hat {j}\)+ \(\frac {2}{\sqrt{14}}\)\(\hat {k}\)
Ques. What are the two properties of vector addition? (2 marks)
Ans. The vector addition is defined as the sum of two or more vectors. Triangle law and parallelogram law are two significant laws related to vector addition. Similarly, the properties of vector addition are as follows:
- Commutative Property
- Associative Property
Ques. What is a Zero Vector, and what does it mean? (2 marks)
Ans. The exception to vectors with direction is a zero vector with no direction. The zero vector, as its name implies, is a vector with zero magnitudes. The zero vector does not point in any direction due to its zero magnitudes. There can only be one vector with a magnitude of zero. It is represented by the number 0 since the length or magnitude is zero. As a result, we call it the zero vector.
Ques. Is it possible to define a vector space using an operation other than conventional addition that nevertheless satisfies the closure property? (2 marks)
Ans. Yes. In a vector space, the 'addition' operation does not have to be the addition of two real numbers. A vector space is defined as a set of nXm matrices. A vector space is defined by continuous functions on a closed interval. There exist vector spaces where exponentiation has substituted addition.
Ques. If the angle between the two vectors is 90 degrees, compute the cross product of two vectors 'k' and 'l' with magnitudes of 7 and 9 units, respectively. (2 marks)
Ans. The dot product of two vectors can be determined using the following formula:
\(\vec {a}\) . \(\vec {b}\) = \(\mid{\vec{a}\mid}\)\(\mid{\vec{b}\mid}\) Cos θ
\(\vec {a}\) . \(\vec {b}\) = 7 x 9 x Cos 90
\(\vec {a}\) . \(\vec {b}\) = 63 x 0
\(\vec {a}\) . \(\vec {b}\) = 0 units
Ques. What are Vector Components? (2 marks)
Ans. The resolution of the vector procedure divides a vector into two or more smaller components. In a vector space, any vector may be broken down into two parts: a horizontal component and a vertical component. The horizontal component is the product of the vector's magnitude and the horizontal angle's cosine. A vector's vertical component is the product of its magnitude and the sine of its horizontal angle.
Ques. Determine the resultant addition vector of the vectors a= (8,13) and b= (12, 15). (2 marks)
Ans. The 'a' and 'b' addition vectors are obtained as
\(\vec {c}\)= \(\vec {a}\)+\(\vec {b}\)
\(\vec {c}\) = (8,13) + (12,15)
\(\vec {c}\) =8+12+13+15
\(\vec {c}\) =(20,28)
Ques. Find the magnitude of vector \(\vec {c}\)=5,12 (2 marks)
Ans. The magnitude of the vector \(\vec {c}\) is calculated as,
\(\mid{\vec{c}\mid}\) = \(\sqrt{x^2+y^2}\)
\(\mid{\vec{c}\mid}\) = \(\sqrt{5^2+2^2}\)
\(\mid{\vec{c}\mid}\) = \(\sqrt{25+144}\)
\(\mid{\vec{c}\mid}\) = \(\sqrt{169}\)
\(\mid{\vec{c}\mid}\) = 13 units
Ques. Find the dot product if the angle of separation between vectors 'a' and 'b' is 60 degrees in one of the vector Mathematics examples if |a| = 5 units and |b| = 10 units. (2 marks)
Ans. The dot product of two vectors can be determined using the following formula:
\(\vec {a}\). \(\vec {b}\) = \(\mid{{a}\mid}\). \(\mid{{b}\mid}\) Cos θ
\(\vec {a}\) .\(\vec {b}\) = 5 x 10 x Cos 60
\(\vec {a}\) .\(\vec {b}\) = 50 x ½
\(\vec {a}\) .\(\vec {b}\) = 25 units
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