Class 12 Mathematics Product of Vectors Notes has been updated according to the latest syllabus of 2080. It means the notes of product of vectors chapter provided in this article contains all the new exercise that has recently been updated. Now you don’t need to go anywhere searching for the notes of this chapter because we are here to serve you.
Chapter – 9
Product of Vectors
The product of vectors can be either the dot product or the cross product:
- Dot Product: The dot product of two vectors $latex \vec{A}$ and $latex \vec{B}$ is a scalar quantity given by $latex \vec{A} \cdot \vec{B}$, which equals $latex |\vec{A}| |\vec{B}| \cos(\theta)$, where $latex \theta$ is the angle between the vectors.
- Cross Product: The cross product of two vectors $latex \vec{A}$ and $latex \vec{B}$ is a vector quantity given by $latex \vec{A} \times \vec{B}$, which equals $latex |\vec{A}| |\vec{B}| \sin(\theta) , \vec{n}$, where $latex \theta$ is the angle between the vectors and $latex \vec{n}$ is a unit vector perpendicular to the plane containing $latex \vec{A}$ and $latex \vec{B}$.
Vector Products of Two Vectors
The vector products of two vectors can be categorized into the dot product and the cross product. The dot product, denoted as $latex \vec{A} \cdot \vec{B}$, results in a scalar and is calculated by $latex |\vec{A}| |\vec{B}| \cos(\theta)$, where $latex \theta$ is the angle between the vectors.
It measures the extent to which one vector projects onto another. The cross product, denoted as $latex \vec{A} \times \vec{B}$, results in a vector orthogonal to both original vectors and is calculated by $latex |\vec{A}| |\vec{B}| \sin(\theta) , \vec{n}$,
where $latex \vec{n}$ is a unit vector perpendicular to the plane containing $latex \vec{A}$ and $latex \vec{B}$. This operation is useful in physics for finding torque and rotational forces.
Class 12 Mathematics Product of Vectors Notes PDF
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The properties of vector products can be divided into those of the dot product and the cross product.
Dot Product Properties
- Commutative Property: The dot product is commutative, meaning $latex \vec{A} \cdot \vec{B} = \vec{B} \cdot \vec{A}$.
- Distributive Property: The dot product is distributive over vector addition, meaning $latex \vec{A} \cdot (\vec{B} + \vec{C}) = \vec{A} \cdot \vec{B} + \vec{A} \cdot \vec{C}$.
- Scalar Multiplication: For any scalar $latex c$, $latex (c\vec{A}) \cdot \vec{B} = c(\vec{A} \cdot \vec{B})$.
- Orthogonality: If $latex \vec{A}$ and $latex \vec{B}$ are orthogonal (perpendicular), then $latex \vec{A} \cdot \vec{B} = 0$.
Cross Product Properties
- Anticommutative Property: The cross product is anticommutative, meaning $latex \vec{A} \times \vec{B} = -(\vec{B} \times \vec{A})$.
- Distributive Property: The cross product is distributive over vector addition, meaning $latex \vec{A} \times (\vec{B} + \vec{C}) = \vec{A} \times \vec{B} + \vec{A} \times \vec{C}$.
- Scalar Multiplication: For any scalar $latex c$, $latex (c\vec{A}) \times \vec{B} = c(\vec{A} \times \vec{B})$.
- Magnitude: The magnitude of the cross product $latex \vec{A} \times \vec{B}$ is $latex |\vec{A} \times \vec{B}| = |\vec{A}| |\vec{B}| \sin(\theta)$, where $latex \theta$ is the angle between $latex \vec{A}$ and $latex \vec{B}$.
- Orthogonality: The resulting vector from the cross product $latex \vec{A} \times \vec{B}$ is orthogonal to both $latex \vec{A}$ and $latex \vec{B}$.
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