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## Chapter – 9

__Product of Vectors__

The product of vectors is a mathematical operation that results in a scalar quantity, representing the magnitude of one vector multiplied by the magnitude of another vector and the cosine of the angle between them.

### Product of vectors

The multiplication operation between two vectors can be performed by the following two ways:

- Scalar product or Dot product
- Vector product or Cross product

The product of two vectors whose resulting value is a scalar is known as the scalar product. On the other hand, if the resulting value is a vector, then the product is known as the vector product.

## Product of Vectors Class 12 Mathematics Notes PDF

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### Scalar Product of Two Vectors

The scalar product of two vectors, also known as the dot product, is a mathematical operation that yields a scalar quantity. It is calculated by multiplying the magnitude of one vector by the magnitude of the other vector and then multiplying this result by the cosine of the angle between the two vectors.

### Angle between two vectors

The angle between two vectors is a measure of the separation or inclination between the directions of the two vectors in a multi-dimensional space. It is calculated using the dot product of the two vectors and their magnitudes.

The formula to find the angle $latex \theta$ between two vectors $latex \vec{A}$ and $latex \vec{B}$ is:

$latex \cos(\theta) = \frac{\vec{A} \cdot \vec{B}}{|\vec{A}| |\vec{B}|}$

where $latex \vec{A} \cdot \vec{B}$ is the dot product of vectors $latex \vec{A}$ and $latex \vec{B}$, and $latex |\vec{A}|$ and $latex |\vec{B}|$ are the magnitudes of vectors $latex \vec{A}$ and $latex \vec{B}$ respectively.

The angle between two vectors can help determine their relationship, such as whether they are parallel, perpendicular, or at an oblique angle to each other. This concept is widely used in physics, engineering, and mathematics to analyze forces, motion, and geometric properties of vectors in various applications.

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