RS Aggarwal Class 12 Vectors and Their Properties PDF

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What are Vectors and Their Properties?

In mathematics, a vector is an element of a vector space, which is a collection of objects that can be added together and multiplied (“scaled”) by numbers, called scalars. Scalars are often taken to be real numbers, but there are also vector spaces with scalar multiplication by complex numbers, rational numbers, or generally any field.

Vectors are often represented by an ordered list of numbers, called coordinates. For example, in two-dimensional space, a vector can be represented by an ordered pair of numbers (x, y). In three-dimensional space, a vector can be represented by an ordered triple of numbers (x, y, z).

The properties of vectors include:

• Addition: Two vectors can be added together component-wise, resulting in a new vector. For example, (x1, y1) + (x2, y2) = (x1 + x2, y1 + y2)
• Subtraction: Two vectors can be subtracted component-wise, resulting in a new vector. For example, (x1, y1) – (x2, y2) = (x1 – x2, y1 – y2)
• Scalar multiplication: A vector can be multiplied by a scalar, resulting in a new vector. For example, k(x, y) = (kx, ky)
• Dot product: Two vectors can be multiplied using the dot product, which is a scalar value. The dot product of vectors A and B is A.B = |A| |B| cos(theta)
• Cross Product: Two vectors can be multiplied using the cross product, which results in a vector that is perpendicular to both A and B. The cross product of vectors A and B is A x B = |A| |B| sin(theta)
• The magnitude of a vector is the distance of the vector from the origin of the vector space. The magnitude of a vector A = |A|
• The unit vector is a vector of magnitude 1.
• A zero vector is a vector that has a magnitude of zero, and it is represented by (0,0,0)

Vectors have wide applications in various fields of physics and engineering as well as in computer science.

CH-22-Vectors-and-There-Properties

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