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Vector Definition
Mathematical representation of physical quantities for which both magnitude and direction can be determined is called a vector. Vector of any physical quantity is represented as a straight line with an arrowhead. In vector definition, the length of the straight line denotes the magnitude of the vector and the arrowhead gives its direction. Any two vectors can be regarded as identical vectors if they have equal magnitude and direction. The best example for a vector is the force applied to an object because both the strength and direction of the applied force affect its action on the object. Rotating or moving a vector around itself will never change its magnitude.
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Vector Math
Vector Math finds a wide range of applications in various domains of Algebra, Geometry, and Physics. As discussed above, a vector is represented as a straight line with an arrowhead. The endpoints of a vector are generally labeled with letters of the English Alphabet in uppercase. Vectors are symbolically represented as the endpoints with an arrowhead or a lower case letter with an arrowhead.
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In the above vector, the region enclosed by a flower bracket indicates the magnitude of the vector and the arrowhead indicates vector direction.
The magnitude of this vector is given as |AB| or |a|. It represents the vector length and is generally calculated with the help of the Pythagorean theorem. Basic Mathematical operations like addition, subtraction, and multiplication can be performed on vectors. However, the division of two vectors is not possible.
Vocabulary of Vectors
Most important terms associated to vectors are:
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Zero Vector : A vector whose magnitude is zero.
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Unit Vector : A vector with magnitude of one unit.
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Position Vector : A vector that denotes the position of a point with respect to its origin.
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Co Initial Vector : Two or more vectors with the same starting point.
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Like and Unlike Vectors : Vectors with the same direction are called like vectors and those with different directions are called unlike vectors.
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Coplanar Vectors : Vectors in the same plane.
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Collinear Vector : Vectors lying on the same straight line.
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Equal Vectors : Two or more vectors with the same magnitude and direction.
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Displacement Vector : A vector indicating displacement of an object from one point to another.
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Negative of a Vector : Negative of any vector is another vector with the same magnitude but opposite direction.
Mathematical Operations on Vector
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Vector Addition
Vector Addition is performed on any two vectors using the triangle law of vector addition. According to this law, the two vectors to be added are represented by two sides of a triangle with the same magnitude and direction. The third side gives the magnitude and direction of the resultant addition vector.
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Vector Subtraction
Considering two vectors a and b. If vector ‘a’ is to be subtracted from vector ‘b’, the negative of vector ‘a’ has to be found and it should be added to vector ‘b’ using triangle law.
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Vector Multiplication
Multiplication of any two vectors is performed by finding their ‘cross product’ or ‘dot product’.
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a b =|a||b| sin θ n̂ where, |a| is the magnitude of vector ‘a’ |b| is the magnitude of vector ‘b’ θ is the angle of separation of two vectors ‘a’ and ‘b’ n̂ is the unit vector representing the direction of multiplication of vectors |
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a . b =|a||b| cos θ where, |a| is the magnitude of vector ‘a’ |b| is the magnitude of vector ‘b’ θ is the angle of separation of two vectors ‘a’ and ‘b’ |
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Cross product of two vectors is a vector quantity. It has both magnitude and direction whereas, the dot product of two vectors has only magnitude and no direction. So, it is a scalar quantity.
Vector Mathematics Examples
1. Find the resultant addition vector of vector a= (8,13) and vector b=(12, 15).
Solution: The addition vector of ‘a’ and ‘b’ obtained as
c = a+b
c = (8, 13) +(12, 15)
c = (8+12)+(13+15)
c = (20, 27)
2. In one of the vector questions, k = (3, 4) and m = (7, 9). Subtract vector ‘k’ from vector ‘m’.
Solution: To subtract vector ‘k’ from vector ‘m’, the negative vector of ‘k’ should be found.
Negative vector of ‘k’ = – k
= – (3, 4)
= ( -3, -4)
The subtraction of vector ‘k’ from vector ‘m’ is given as:
m – k = m + (-k)
= (7, 9) + (-3, -4)
= (7 – 3), (9 – 4)
= (4, 5)
3. Determine the magnitude of vector c = (5, 12)
Solution: Magnitude of vector ‘c’ is calculated as,
|c| = [sqrt{x^{2} + y^{2}}]
|c| = [sqrt{5^{2} + 12^{2}}]
|c| = [sqrt{25 + 144}]
|c| = [sqrt{169}]
|c| = 13 units
4. In one of the vector Mathematics examples, if |a| = 5 units and |b| = 10 units, find the dot product if the angle of separation between vector ‘a’ and ‘b’ is 60o.
Solution: Dot product of two vectors can be calculated as:
[a cdot b = |a||b| cos theta]
[a cdot b = 5times 10times cos 60^{o}]
[a cdot b = 50times frac{1}{2}]
[a cdot b = 25 text{units}]
5. Compute cross product of 2 vectors ‘k’ and ‘l’ whose magnitudes are 7 units and 9 units respectively if the angle between the two vectors is 90o.
Solution: Dot product of two vectors can be calculated as:
[a cdot b = |a||b| cos theta]
[a cdot b = 7times 9times cos 90^{o}]
[a cdot b = 63 times 0]
[a cdot b = 0 text{units}]
Fun Facts
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Any geometric object which has both magnitude and direction is called a Euclidean Vector.
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Matrices can also be used with the help of vector definition. Any matrix with a single row or a single column is termed as row vector or column vector respectively.