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Vector Product of Two Vectors | Physics Class 11 - NEET PDF Download
The cross product, area product or the vector product of two vectors is a binary operation on two vectors in three-dimensional spaces. It is denoted by ×. The cross product of two vectors is a vector. Let us consider two vectors denoted as. Let the product (also a vector) of these two vectors be denoted as.
Magnitude of the vector product
The magnitude of the vector product is given as,
Where a and b are the magnitudes of the vector and Ɵ is the angle between these two vectors. From the figure, we can see that there are two angles between any two vectors, that is, Ɵ and (360° – Ɵ). In this rule, we always consider the smaller angle that is less than 180°.
Direction of the vector product
The right-hand thumb rule is used in which we curl up the fingers of right hand around a line perpendicular to the plane of the vectors a and b and the curl the fingers in the direction from a to b, then the stretched thumb points in the direction of c.
Commutative property
Unlike the scalar product, cross product of two vectors is not commutative in nature. Mathematically, for scalar products
But for vector products
As we know, the magnitude of both the cross products a × b and b × a is the same and is given by absinθ; but the curling of the right-hand fingers in case of a × b is from a to b, whereas in case of (b × a) it is from b to a, as per which, the two vectors are in opposite directions.
Mathematically,
Distributive property
Like the scalar product, vector product of two vectors is also distributive with respect to vector addition. Mathematically,
In order to deal with the vector product of any two vectors, we need to know the vector product of two elementary vectors.
Similarly, for the unit vectors following results hold good,
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FAQs on Vector Product of Two Vectors - Physics Class 11 - NEET
| 1. What is the vector product of two vectors? |  |
The vector product, also known as the cross product, is an operation that takes two vectors as input and produces a third vector that is perpendicular to both of the input vectors. It is denoted by the symbol ×.
| 2. How is the vector product calculated? | |
To calculate the vector product of two vectors, you can use the following formula:
A × B = |A| |B| sin(θ) n
where A and B are the input vectors, |A| and |B| are their magnitudes, θ is the angle between them, and n is a unit vector perpendicular to both A and B.
| 3. What is the significance of the angle between the two vectors in the vector product? | |
The angle between the two vectors plays a crucial role in determining the magnitude and direction of the resulting vector. The magnitude of the vector product is given by |A| |B| sin(θ), where θ is the angle between A and B. If the vectors are parallel or anti-parallel, the angle is 0 or 180 degrees, resulting in a vector with zero magnitude.
| 4. How can the direction of the vector product be determined? | |
The direction of the vector product can be determined using the right-hand rule. If you align your right hand's fingers along vector A and curl them towards vector B, then your thumb will point in the direction of the resulting vector. Alternatively, you can use the cross product rule, which states that the direction of A × B is perpendicular to both A and B.
| 5. What are some applications of the vector product in physics and engineering? | |
The vector product has various applications in physics and engineering. It is used in calculating torque, magnetic fields, angular momentum, and fluid flow. For example, in electromagnetism, the vector product is used to determine the direction of the magnetic field produced by a current-carrying wire. In mechanics, it is employed to calculate the rotational effects of forces on objects.
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