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What is vector (in physics)?
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In physics, a vector is a quantity that has both magnitude (size) and direction. Vectors are used to describe anything that depends on direction as well as amount, for example:
- Displacement, velocity, acceleration
- Force, momentum, electric and magnetic fields
How vectors are represented
- Geometrically: an arrow whose length is the magnitude and whose orientation is the direction.
- Algebraically: by components in a coordinate system, e.g. v = (v_x, v_y, v_z) or v = v_x i + v_y j + v_z k, where i, j, k are unit basis vectors.
- Magnitude: |v| = sqrt(v_x^2 + v_y^2 + v_z^2).
Basic operations
- Addition: vector sum follows the triangle/parallelogram rule; components add: a + b = (a_x + b_x, a_y + b_y, a_z + b_z). Addition is commutative.
- Scalar multiplication: c a = (c a_x, c a_y, c a_z) scales magnitude and possibly reverses direction if c < 0.
- Dot product (scalar product): a · b = |a||b| cosθ = a_x b_x + a_y b_y + a_z b_z. Result is a scalar.
- Cross product (vector product, in 3D): a × b = vector of magnitude |a||b| sinθ perpendicular to both a and b (direction given by right-hand rule). Compute via the determinant formula or component expressions.
Other useful notions
- Unit vector: a vector of magnitude 1 used to indicate direction (e.g. ê = v/|v|).
- Zero vector: the vector with zero magnitude and undefined direction.
- Free vs bound vectors: free vectors can be moved parallelly without changing them (common in mechanics); bound (position) vectors are attached to a point.
- Vector field: a vector assigned to every point in space, e.g. velocity field of a fluid or an electric field E(r).
Contrast with scalars
- Scalars have magnitude only (temperature, mass, time) and do not carry direction.
Example (2D)
If a = (3, 4) and b = (−1, 2):
- a + b = (2, 6)
- |a| = 5
- a · b = 3(−1) + 4(2) = 5
- a × b (z-component) = 3·2 − 4·(−1) = 6 + 4 = 10 (gives magnitude and sign for perpendicular direction)
That is the basic idea: vectors let you treat direction-dependent physical quantities rigorously and compute results using component methods or geometric rules.
- Displacement, velocity, acceleration
- Force, momentum, electric and magnetic fields
How vectors are represented
- Geometrically: an arrow whose length is the magnitude and whose orientation is the direction.
- Algebraically: by components in a coordinate system, e.g. v = (v_x, v_y, v_z) or v = v_x i + v_y j + v_z k, where i, j, k are unit basis vectors.
- Magnitude: |v| = sqrt(v_x^2 + v_y^2 + v_z^2).
Basic operations
- Addition: vector sum follows the triangle/parallelogram rule; components add: a + b = (a_x + b_x, a_y + b_y, a_z + b_z). Addition is commutative.
- Scalar multiplication: c a = (c a_x, c a_y, c a_z) scales magnitude and possibly reverses direction if c < 0.
- Dot product (scalar product): a · b = |a||b| cosθ = a_x b_x + a_y b_y + a_z b_z. Result is a scalar.
- Cross product (vector product, in 3D): a × b = vector of magnitude |a||b| sinθ perpendicular to both a and b (direction given by right-hand rule). Compute via the determinant formula or component expressions.
Other useful notions
- Unit vector: a vector of magnitude 1 used to indicate direction (e.g. ê = v/|v|).
- Zero vector: the vector with zero magnitude and undefined direction.
- Free vs bound vectors: free vectors can be moved parallelly without changing them (common in mechanics); bound (position) vectors are attached to a point.
- Vector field: a vector assigned to every point in space, e.g. velocity field of a fluid or an electric field E(r).
Contrast with scalars
- Scalars have magnitude only (temperature, mass, time) and do not carry direction.
Example (2D)
If a = (3, 4) and b = (−1, 2):
- a + b = (2, 6)
- |a| = 5
- a · b = 3(−1) + 4(2) = 5
- a × b (z-component) = 3·2 − 4·(−1) = 6 + 4 = 10 (gives magnitude and sign for perpendicular direction)
That is the basic idea: vectors let you treat direction-dependent physical quantities rigorously and compute results using component methods or geometric rules.
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