Name: cross product direction and magnitude
Uploaded: 2023-09-22T11:49:20.000Z
Duration: 1 min 25 s
Channel: JoVE

Cross-product is a product of two different vectors whose resultant is a third vector.
The magnitude of the cross-product is obtained by multiplying the magnitude of both the vectors and the sine of the angle between them.
Here, the magnitude of the cross-product can be multiplied with a unit vector that specifies the direction of the resultant vector.
The right-hand rule gives the direction of the resultant vector. If the curled fingers represent the direction from vector A to vector B,&#160; then the direction of the thumb represents the direction of the resultant vector.
The resultant vector is always perpendicular to the plane containing vectors A and B.
The cross-product of vectors is non-commutative. The resultant of the cross-product of vector A with vector B is equal in magnitude but opposite in direction to that of the cross-product of vector B with vector A.
However, the cross-product obeys the associative and distributive laws.

cross product

The cross product is a fundamental concept in vector algebra that is a vector operation on two different vectors to obtain a third vector. Unlike the scalar product, the cross product results in a vector quantity perpendicular to both the original vectors.
The magnitude of the cross product is obtained by multiplying the magnitude of both the vectors and the sine of the angle between them. This means that a larger angle between the vectors will lead to a greater magnitude of the cross product.
<img alt="C=ABsinθ equation representing trigonometric vector relationships in a mathematical diagram." src="/files/ftp_upload/14251/14251_Equation_1v3.svg" style="text-align: center; height: 11px;" data-altupdatedat="1761830439000" data-alt="Equation 1">
The direction of the resultant vector is determined by using the right-hand rule. If we curl the fingers of the right hand from vector A to vector B, then the direction of the thumb represents the direction of the resultant vector. In other words, the direction of the cross product is perpendicular to the plane containing vectors A and B.
It is important to note that the cross product of vectors is non-commutative. That is, the resultant vector for the cross product of vector A with vector B is equal in magnitude but opposite in direction to that of the cross product of vector B with vector A. However, the cross product obeys the associative law and distributive laws of addition. This means that the cross product of a sum of vectors equals the sum of the cross products of each vector.
In conclusion, the cross product is a fundamental concept in vector algebra that has widespread applications in physics and engineering, such as calculating the moment of a force about a point, calculating the torque of a force about an axis, and calculating the angular momentum of a body about an axis.

Cross Product

The cross product is a fundamental concept in vector algebra that is a vector operation on two different vectors to obtain a third vector. Unlike the ...

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cross product direction and magnitude

The cross product is a vector operation in vector algebra that creates a third vector from two input vectors. Unlike the scalar product, it does not give a single number. The result is a vector perpendicular to both original vectors.

The size, or magnitude, of the cross product depends on the magnitudes of both vectors and the sine of the angle between them. A larger angle between the vectors gives a larger cross product magnitude. The direction is found with the right-hand rule. If the fingers of the right hand curl from vector A to vector B, the thumb shows the direction of the result.

The cross product points perpendicular to the plane containing vectors A and B. It is also non-commutative. That means A cross B has the same magnitude as B cross A, but the direction is opposite.

The cross product still follows the associative law and the distributive law of addition. So, the cross product of a sum of vectors equals the sum of the cross products of each vector. This idea is useful in physics and engineering. It is used to calculate the moment of a force about a point, torque about an axis, and angular momentum about an axis.

Cross Product Direction and Magnitude

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The cross product is a vector operation in vector algebra that creates a third vector from two input vectors. Unlike the scalar product, it does not give a single number. The result is a vector perpendicular to both original vectors. The size, or magnitude, of the cross product depends on the magnitudes of both vectors and the sine of the angle between them. A larger angle between the vectors gives a larger cross product magnitude. The direction is found with the right-hand rule. If the...

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Learn how the cross product creates a perpendicular vector, how to use the right-hand rule, and why its size depends on angle, torque, and angular momentum.

Cross Product Direction and Magnitude | JoVE K12

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Cross Product Direction and Magnitude

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