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Physics: Problems and Solutions

Vector

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In mathematics and physics, a vector is a quantity with both magnitude and direction. Vectors are commonly used (usually unknowingly) in everyday life; for instance, "five miles west" is a vector. Common vectors include position, velocity, and acceleration. Vectors are crucial in physics, as well as some mathematical fields. Common vectors include force, momentum, acceleration, velocity, and position.

Vectors can be denoted many different ways. For example, the vector \vec{\mathbf{v}} with a magnitude of three in the x-direction, two in the y-direction and four in the z-direction can be denoted with ordered set notation, matrix notation, or unit vector notation:

\vec{\mathbf{v}} = (3, 2, 4) = 
\begin{bmatrix}
3\\
2\\
4\\ 
\end{bmatrix}
 = 3\mathbf{\hat{i}} + 2\mathbf{\hat{j}} + 4\mathbf{\hat{k}}

Vectors are first order tensors.

PseudovectorsEdit

Pseudovectors are derived as the cross product of two other vectors. In rotational systems, pseudovectors such as torque, angular momentum, and angular velocity point in a direction perpendicular to the plane of rotation as directed by the right-hand rule.

Right-hand ruleEdit

The right hand rule is a memory device used for deciding the direction of a vector attained by taking a cross product. In the case of rotational systems, the direction of the vector is the same as the direction the right thumb points when the fingers are curled.

Vector operationsEdit

  • Vector addition: Vectors can be added by simply adding the components in each direction.
  • Scalar multiplication: Vectors can be multiplied by scalars by multiplying each component of the vector by said scalar.
  • Vector multiplication: This can be done in two ways, yielding either the dot or cross product.

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