What is Vector?
A vector is an object with both magnitude and direction. A vector can be visualized geometrically as a guided line segment with an arrow indicating the direction and a length equal to the magnitude of the vector. If two vectors have the same magnitude and direction, they are the same.
Vector Operations
Vector algebra refers to the algebraic operations in vector calculus that are defined for a vector space and then applied globally to a vector field. The following are the basic vector algebraic operations:
Vector addition
Vector subtraction
Scalar multiplication
Dot product
Cross product
The vector calculation rules of various vector operations are discussed below.
Vector Addition
The resultant vector can be computed by adding two vectors together. Vector addition is the method of combining two or more vectors.
Assume that a and b are not inherently identical vectors and that their magnitudes and directions can vary. The resultant of a and b equals
a + b = (a1 + b1) e1 + (a2 + b2) e2 +(a3 + b3) e3
Place the tail of the arrow b at the head of the arrow a, and then draw an arrow from the tail of a to the head of b to show the addition graphically. The vector a + b is represented by the new arrow drawn, as shown below.
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Because a and b are the sides of a parallelogram, and a + b is one of the diagonals, this vector addition method is sometimes referred to as the parallelogram rule. This point would also be the base point of a + b if a and b are connected vectors with the same base point.
Associative Law of Vector Addition
The associative law of vectors states that regardless of the order or grouping in which vectors are grouped, the number of vectors remains the same.
To prove vector addition is associative consider three vectors \[\bar{A}\], \[\bar{B}\] and \[\bar{C}\] .
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To obtain the resultant vector apply head to tail rule that is (\[\bar{A}\] + \[\bar{B}\]) and (\[\bar{B}\] + \[\bar{C}\])
From the diagram, the resultant vector will be
\[\bar{OR}\] = \[\bar{OP}\] + \[\bar{PR}\]
\[\bar{R}\] = \[\bar{A}\] + (\[\bar{B}\] + \[\bar{C}\]) ………….(1)
and
\[\bar{OR}\] = \[\bar{OQ}\] + \[\bar{QR}\]
\[\bar{R}\] = (\[\bar{A}\] + \[\bar{B}\]) + \[\bar{C}\] ………….(2)
Now equating equations (1) and (2)
\[\bar{A}\] + (\[\bar{B}\] + \[\bar{C}\]) = (\[\bar{A}\] + \[\bar{B}\]) + \[\bar{C}\]
This is known as the associative law of vector addition.
Vector Subtraction
The difference of a vector can be computed by subtracting one vector from another.
Assume that a and b are not inherently identical vectors and that their magnitudes and directions can vary. The difference of the vectors a and b is
a - b = (a1 - b1) e1 + (a2 - b2) e2 +(a3 - b3) e3
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The geometric representation of subtracting two vectors is as follows: to subtract b from a, align the tails of a and b at the same location, and then draw an arrow from the head of b to the head of a. The vector (-b) + a is represented by this new arrow, with (-b) being the inverse of b and
(-b) + a = a - b.
Vector Multiplication
Vector multiplication is a term that refers to one of many methods for multiplying two or more vectors by themselves.
The various vector multiplication rules are as follows:
Dot Product
The dot product, also known as the scalar product, is a mathematical operation that returns a scalar quantity from two vectors. The product of the magnitudes of the two vectors and the cosine of the angle between them is known as the dot product of two vectors. It is also known as the product of the first vector's projection onto the second vector and the magnitude of the second vector. The vector multiplication rules for the dot product is as follows:
A . B =|A| |B| Cos θ
Cross Product
The cross product, also known as the vector product, is a binary operation that produces another vector from two vectors. The vector perpendicular to the plane determined by the cross product of two vectors in 3-space is defined as the vector whose magnitude is the product of the magnitudes of the two vectors and the sine of the angle between the two vectors. As a result, if n̂ is the perpendicular unit vector to the plane defined by vectors A and B as
A x B =|A| |B| Sin θ n̂
Triple Product
A triple product is a three-dimensional vector product, typically Euclidean vectors. The scalar-valued scalar triple product and, less often, the vector-valued vector triple product are both referred to as triple products.
Scalar Triple Product
The scalar triple product is defined as the dot product of one vector with the cross product of the other two. It is also known as the mixed product, box product, or triple scalar product. Scalar triple product is given as
a . (b x c) = det (a, b, c)
Vector Triple Product
The cross product of one vector with the cross product of the other two is known as the vector triple product.
The vector triple product is given as follows:
a x (b x c) = (a . c)b - (a . b)c
Conclusion
The laws of elementary algebra are extended to vectors by vector operations. Addition, subtraction, and three forms of multiplication are among them. The diagonal of the parallelogram built with the two initial vectors as sides are the sum of two vectors, which is defined as a third vector. When a vector is multiplied by a positive scalar, its magnitude is multiplied by the scalar, and its direction is unchanged; however, if the scalar is negative, the direction is reversed. The dot product, written a b, and the cross product, written a b, are the results of multiplying a vector by another vector b. For vector addition and the dot product, the associative and commutative laws apply. The cross product is not commutative, but it is associative.
FAQs on Vector Operations
1. What are the fundamental operations in vector algebra?
The fundamental operations in vector algebra, as per the CBSE syllabus, involve combining or transforming vectors. The primary operations are:
- Vector Addition and Subtraction: Combining two or more vectors to find a resultant vector.
- Scalar Multiplication: Scaling a vector by a constant, which changes its magnitude and potentially reverses its direction.
- Scalar (Dot) Product: An operation between two vectors that results in a scalar quantity.
- Vector (Cross) Product: An operation between two vectors that results in a new vector perpendicular to the plane containing the original two.
2. How is vector addition performed using the triangle and parallelogram laws?
Both laws provide a geometric way to add two vectors. According to the Triangle Law of Vector Addition, if two vectors are represented as two sides of a triangle in sequence (head to tail), the third side of the triangle, drawn from the starting point of the first vector to the endpoint of the second, represents the resultant vector. The Parallelogram Law of Vector Addition states that if two vectors are represented by the two adjacent sides of a parallelogram drawn from a common point, their resultant is represented by the diagonal of the parallelogram that passes through that same common point.
3. What is the difference between the scalar (dot) product and the vector (cross) product?
The key difference lies in their output and geometric meaning. The scalar (dot) product of two vectors, a and b, results in a scalar value and is calculated as |a||b|cosθ. It represents the projection of one vector onto another. In contrast, the vector (cross) product, a × b, results in a new vector that is perpendicular to both a and b. Its magnitude is |a||b|sinθ, which represents the area of the parallelogram formed by the two vectors.
4. How do you find the projection of one vector onto another?
The projection of one vector onto another is a practical application of the dot product. To find the projection of vector a onto vector b, you use the formula:
Projection = (a ⋅ b̂) = (a ⋅ b) / |b|.
This formula calculates the scalar component of vector a in the direction of vector b. It essentially tells you how much of vector a lies along the line defined by vector b.
5. What is scalar multiplication and how does it affect a vector?
Scalar multiplication is the operation of multiplying a vector by a scalar (a real number). This operation scales the vector's magnitude and can change its direction. If a vector v is multiplied by a scalar k, the new vector kv has a magnitude of |k| times the magnitude of v. If k is positive, the direction remains the same. If k is negative, the direction is exactly reversed.
6. Why is a unit vector essential for defining direction in vector operations?
A unit vector is essential because its magnitude is exactly 1, meaning its sole purpose is to specify a direction in space without altering the magnitude of any vector it is multiplied with. This is fundamentally important for defining coordinate systems (like the standard basis vectors î, ĵ, k̂) and for expressing the direction of a resultant vector. For instance, in the cross product formula a × b = |a||b|sinθ n̂, the unit vector n̂ is crucial for defining the direction of the resulting perpendicular vector.
7. Why is the cross product of two collinear vectors the zero vector?
The cross product of two vectors is zero if they are collinear (i.e., parallel or anti-parallel). This is because the magnitude of the cross product of vectors a and b is given by |a||b|sinθ, where θ is the angle between them. For collinear vectors, the angle θ is either 0° or 180°. Since sin(0°) = 0 and sin(180°) = 0, the magnitude of the resultant vector is always zero. A vector with zero magnitude is defined as the zero vector (0).
8. What is the geometric interpretation of the scalar triple product?
The geometric interpretation of the scalar triple product of three vectors, a, b, and c, is a measure of volume. The absolute value of the scalar triple product, |a ⋅ (b × c)|, represents the volume of the parallelepiped whose adjacent edges are defined by the three vectors. If the scalar triple product is zero, it implies the three vectors are coplanar, meaning they lie on the same plane and the volume they form is zero.
