About vector formulas physics
Here are some of the commonly used vector formulas in physics:
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The magnitude of a vector: The magnitude (or length) of a vector is given by:
|a| = √(a?² + a?² + a?²)
where a?, a?, and a? are the components of the vector in the x, y, and z directions, respectively.
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Unit vector: A unit vector is a vector with a magnitude of 1. To find the unit vector in the direction of a vector a, we divide a by its magnitude:
? = a / |a|
where ? is the unit vector in the direction of a.
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Dot product: The dot product (or scalar product) of two vectors a and b is given by:
a · b = |a| |b| cos(θ)
where θ is the angle between the two vectors. This formula is often used to find the angle between two vectors.
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Cross product: The cross product (or vector product) of two vectors a and b is given by:
a x b = |a| |b| sin(θ) n
where n is a unit vector perpendicular to both a and b, and θ is the angle between the two vectors. This formula is often used to find the direction of a vector perpendicular to two other vectors.
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Component form: A vector a can be written in component form as:
a = a?i + a?j + a?k
where i, j, and k are unit vectors in the x, y, and z directions, respectively.
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Projection: The projection of a vector a onto a vector b is given by:
proj_b a = (a · b / |b|²) b
where · represents the dot product.
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Scalar projection: The scalar projection of a vector an onto a vector b is given by:
|proj_b a| = |a| cos(θ)
where θ is the angle between the two vectors.
These are just a few of the many vector formulas used in physics. Understanding these formulas and how to use them is essential for solving problems in mechanics, electromagnetism, and other areas of physics.