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The sine function is used in the cross product because the magnitude of the cross product of two vectors is determined by the area of the parallelogram formed by those vectors. This area is calculated as the product of the magnitudes of the vectors and the sine of the angle between them. Specifically, the formula for the cross product (\mathbf{A} \times \mathbf{B}) includes (|\mathbf{A}||\mathbf{B}|\sin(\theta)), where (\theta) is the angle between the vectors, capturing the component of one vector that is perpendicular to the other. Thus, the sine function accounts for the directional aspect of the vectors in determining the resultant vector's magnitude and orientation.
What else can I help you with?
Why in dot product you use cos and in vector product sin?
We use the dot product cos and in vector we use the vector product sin because of the trigonometric triangle.
Why you use cosine theta with cross product?
Normally you use sine theta with the cross product and cos theta with the vector product, so that the cross product of parallel vectors is zero while the dot product of vectors at right angles is zero.
How angle between two vectors is found using cross product?
A x B = |A| |B| sin[theta]
What are the applications of cross product and dot product?
Cross product tests for parallelism and Dot product tests for perpendicularity. Cross and Dot products are used in applications involving angles between vectors. For example given two vectors A and B; The parallel product is AxB= |AB|sin(AB). If AXB=|AB|sin(AB)=0 then Angle (AB) is an even multiple of 90 degrees. This is considered a parallel condition. Cross product tests for parallelism. The perpendicular product is A.B= -|AB|cos(AB) If A.B = -|AB|cos(AB) = 0 then Angle (AB) is an odd multiple of 90 degrees. This is considered a perpendicular condition. Dot product tests for perpendicular.
When does the magnitude of dot product and cross product of vectors is equal?
If x is the angle between the two vectors then the magnitudes are equal if cos(x) = sin(x). That is, when x = pi/4 radians.
Why we use sine angle in cross product while cos angle in dot product?
Because in dot product we take projection fashion and that is why we used cos and similar in cross product we used sin
Why in dot product you use cos and in vector product sin?
We use the dot product cos and in vector we use the vector product sin because of the trigonometric triangle.
Why you use cosine theta with cross product?
Normally you use sine theta with the cross product and cos theta with the vector product, so that the cross product of parallel vectors is zero while the dot product of vectors at right angles is zero.
How angle between two vectors is found using cross product?
A x B = |A| |B| sin[theta]
When is a cross product zero?
When the component vectors have equal or opposite directions (sin(Θ) = 0) i.e. the vectors are parallel.
What are the applications of cross product and dot product?
Cross product tests for parallelism and Dot product tests for perpendicularity. Cross and Dot products are used in applications involving angles between vectors. For example given two vectors A and B; The parallel product is AxB= |AB|sin(AB). If AXB=|AB|sin(AB)=0 then Angle (AB) is an even multiple of 90 degrees. This is considered a parallel condition. Cross product tests for parallelism. The perpendicular product is A.B= -|AB|cos(AB) If A.B = -|AB|cos(AB) = 0 then Angle (AB) is an odd multiple of 90 degrees. This is considered a perpendicular condition. Dot product tests for perpendicular.
When does the magnitude of dot product and cross product of vectors is equal?
If x is the angle between the two vectors then the magnitudes are equal if cos(x) = sin(x). That is, when x = pi/4 radians.
How do you use the right hand rule for the cross product in vector mathematics?
To use the right hand rule for the cross product in vector mathematics, align your right hand fingers in the direction of the first vector, then curl them towards the second vector. Your thumb will point in the direction of the resulting cross product vector.
How do you multiply vectors in 3D?
To multiply two vectors in 3D, you can use the dot product or the cross product. The dot product results in a scalar quantity, while the cross product produces a new vector that is perpendicular to the original two vectors.
When cross and dot product equal what is angle between A and B?
A · B = |A| |B| cos(Θ)A x B = |A| |B| sin(Θ)If [ A · B = A x B ] then cos(Θ) = sin(Θ).Θ = 45°
What is the cross product of a vector itself?
0 is a cross product of a vector itself
If you get tattoo is it a sin or if you get a cross on your arm is it a sin?
It is only a sin if it is against your religion. Tatoos are a great way to express oneself.
