12th Mathematics (Arts & Science) Part 1 Chapter 5 Solution (Digest) Maharashtra state board

Chapter 5 Vectors

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Vectors are fundamental objects in mathematics and physics, used to represent quantities that have both magnitude and direction. They are essential in various fields, including geometry, physics, engineering, and computer science. Here's a comprehensive overview of vectors in mathematics:

Definition and Representation

1. Definition:

   • A vector is a mathematical entity that has both magnitude (length) and direction.
   • It can be represented in a coordinate system by an ordered list of numbers, known as components.

2. Representation:

   • Graphically: A vector is represented by an arrow. The length of the arrow indicates the magnitude, and the direction of the arrow shows the direction of the vector.
   • Algebraically: In a Cartesian coordinate system, a vector can be represented as an ordered pair or triplet. For example, in two dimensions, a vector $\mathbf{v}$ can be represented as $\mathbf{v} = (v_x, v_y)$. In three dimensions, it can be $\mathbf{v} = (v_x, v_y, v_z)$.

Basic Operations

1. Addition and Subtraction:

   • Addition: The sum of two vectors $\mathbf{u} = (u_x, u_y)$ and $\mathbf{v} = (v_x, v_y)$ is given by $\mathbf{u} + \mathbf{v} = (u_x + v_x, u_y + v_y)$.
   • Subtraction: The difference between two vectors $\mathbf{u}$ and $\mathbf{v}$ is $\mathbf{u} - \mathbf{v} = (u_x - v_x, u_y - v_y)$.

2. Scalar Multiplication:

   • Multiplying a vector $\mathbf{v} = (v_x, v_y)$ by a scalar $k$ scales the vector's magnitude by $k$, resulting in $k\mathbf{v} = (kv_x, kv_y)$.

3. Magnitude (Norm):

   • The magnitude (or length) of a vector $\mathbf{v} = (v_x, v_y)$ is given by $|\mathbf{v}| = \sqrt{v_x^2 + v_y^2}$ in two dimensions, and $|\mathbf{v}| = \sqrt{v_x^2 + v_y^2 + v_z^2}$ in three dimensions.

4. Direction:

   • The direction of a vector can be described by the angle it makes with a reference axis, often calculated using trigonometric functions.

Advanced Operations

1. Dot Product (Scalar Product):

   • The dot product of two vectors $\mathbf{u} = (u_x, u_y, u_z)$ and $\mathbf{v} = (v_x, v_y, v_z)$ is given by $\mathbf{u} \cdot \mathbf{v} = u_x v_x + u_y v_y + u_z v_z$.
   • It results in a scalar and is useful for finding the angle between two vectors and in projections.

2. Cross Product (Vector Product):

   • The cross product of two vectors $\mathbf{u}$ and $\mathbf{v}$ in three dimensions is a vector $\mathbf{u} \times \mathbf{v}$ perpendicular to both $\mathbf{u}$ and $\mathbf{v}$, with magnitude $|\mathbf{u} \times \mathbf{v}| = |\mathbf{u}||\mathbf{v}|\sin(\theta)$, where $\theta$ is the angle between $\mathbf{u}$ and $\mathbf{v}$.
   • The direction is given by the right-hand rule.

3. Vector Projections:

   • The projection of vector $\mathbf{u}$ onto vector $\mathbf{v}$ is given by $\text{proj}_{\mathbf{v}} \mathbf{u} = \left( \frac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{v}|^2} \right) \mathbf{v}$.

Applications

1. Physics:

   • Vectors are used to represent physical quantities such as displacement, velocity, acceleration, and force.

2. Engineering:

   • Vectors are essential in statics and dynamics for analyzing forces and motion.

3. Computer Graphics:

   • Vectors are used to represent points, directions, and transformations in 2D and 3D graphics.

4. Mathematics:

   • Vectors are fundamental in linear algebra, where they represent points in vector spaces and are used in solving systems of linear equations.

Conclusion

Vectors are versatile and powerful tools in mathematics and its applications. Understanding their properties and operations allows for the analysis and solution of a wide range of problems in various scientific and engineering disciplines.