The magnitude of a vector is calculated by taking the square root of the sum of the squares of its components. In other words, if the vector is represented as (x, y), the magnitude can be found using the formula √(x² + y²).
A scalar is a quantity that has only magnitude, while a vector has both magnitude and direction. Scalars are represented by a single number, while vectors are represented by multiple numbers or components.
The direction of a vector can be found using the formula tan⁻¹(y/x), where y is the vertical component and x is the horizontal component. This will give the direction in radians, which can be converted to degrees if desired.
Yes, vectors can be added or subtracted by adding or subtracting their corresponding components. For example, (x₁, y₁) + (x₂, y₂) = (x₁ + x₂, y₁ + y₂). This is known as the vector addition or subtraction rule.
The dot product of two vectors is calculated by multiplying their corresponding components and then adding the results. For example, (x₁, y₁) · (x₂, y₂) = (x₁x₂ + y₁y₂). This can also be written as ||A|| ||B|| cos(θ), where ||A|| and ||B|| are the magnitudes of the two vectors and θ is the angle between them.