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What is vector in science?
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A vector (in science and mathematics) is a quantity that has both magnitude (size) and direction. It contrasts with a scalar, which has only magnitude.
Key points
- Common physical examples: displacement, velocity, acceleration, force, momentum.
- Representation: drawn as an arrow (length = magnitude, arrowhead = direction); written as v or →v or bold v.
- Components: in coordinate form a = (a_x, a_y, a_z) in 3D.
- Magnitude (length): |a| = sqrt(a_x^2 + a_y^2 + a_z^2).
- Unit vector: â = a / |a| (direction with magnitude 1).
- Operations:
- Addition/subtraction: add components (graphically: tip-to-tail or parallelogram rule).
- Scalar multiplication: scales magnitude, may reverse direction if scalar is negative.
- Dot product (scalar product): a · b = |a||b| cosθ = a_x b_x + a_y b_y + a_z b_z (gives a scalar).
- Cross product (in 3D): a × b = vector perpendicular to both; magnitude |a||b| sinθ and direction given by right-hand rule (gives a vector).
Mathematically, vectors are elements of a vector space (e.g., R^n) and follow rules of vector addition and scalar multiplication.
If you want, I can show examples (e.g., adding two velocity vectors), draw a diagram, or explain dot/cross product in more detail.
Key points
- Common physical examples: displacement, velocity, acceleration, force, momentum.
- Representation: drawn as an arrow (length = magnitude, arrowhead = direction); written as v or →v or bold v.
- Components: in coordinate form a = (a_x, a_y, a_z) in 3D.
- Magnitude (length): |a| = sqrt(a_x^2 + a_y^2 + a_z^2).
- Unit vector: â = a / |a| (direction with magnitude 1).
- Operations:
- Addition/subtraction: add components (graphically: tip-to-tail or parallelogram rule).
- Scalar multiplication: scales magnitude, may reverse direction if scalar is negative.
- Dot product (scalar product): a · b = |a||b| cosθ = a_x b_x + a_y b_y + a_z b_z (gives a scalar).
- Cross product (in 3D): a × b = vector perpendicular to both; magnitude |a||b| sinθ and direction given by right-hand rule (gives a vector).
Mathematically, vectors are elements of a vector space (e.g., R^n) and follow rules of vector addition and scalar multiplication.
If you want, I can show examples (e.g., adding two velocity vectors), draw a diagram, or explain dot/cross product in more detail.
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