What are vectors?
vectors are like numbers, but they also have a direction.
The best first place to try on general questions like this is google.
You will find many examples, illustrations, discussions, and videos.
vectors are quantities that have both magnitude and direction
forces , field strengths , velocities , displacements , etc.
The way I usually bring it up is I can tell my car's speed with a speedometer but to get the velocity I also need a compass.
Vectors are mathematical entities that represent both magnitude (or size) and direction. They are commonly used to describe quantities that have both of these properties, such as displacement, velocity, and force.
To understand what vectors are, it helps to know the following:
1. Magnitude: The magnitude of a vector represents its size or length. It is a scalar value (a number) that describes the quantity being represented. For example, if we have a vector representing the displacement of an object, its magnitude could be the distance traveled by the object.
2. Direction: The direction of a vector describes where it is pointing. It is typically represented by an arrow, with the arrowhead indicating the direction. For example, if we have a vector representing the velocity of an object, its direction could be the direction in which the object is moving.
Vectors can be represented in various ways, such as as arrows, ordered pairs, or column matrices. They can be added, subtracted, multiplied, and divided just like regular numbers, but there are specific rules and operations for vector operations.
To calculate with vectors, you need to understand the basic vector operations:
1. Addition: To add two vectors, you simply add their corresponding components. For example, if you have a vector (2, 3) and another vector (5, -1), you can add them to get (7, 2).
2. Subtraction: To subtract one vector from another, you subtract the corresponding components. For example, if you have a vector (2, 3) and another vector (5, -1), you can subtract them to get (-3, 4).
3. Scalar multiplication: To multiply a vector by a scalar (a number), you multiply each component of the vector by the scalar. For example, if you have a vector (2, 3) and multiply it by 2, you get (4, 6).
4. Dot product: The dot product of two vectors is a scalar value that represents the projection of one vector onto the other. It is calculated by multiplying the corresponding components of the vectors and then summing them up. For example, the dot product of the vectors (2, 3) and (5, -1) is (2*5) + (3*-1) = 7.
5. Cross product: The cross product of two vectors is a vector that is perpendicular to both of the original vectors. It is calculated using a more advanced formula and results in a vector that is orthogonal (perpendicular) to both input vectors.
Understanding and working with vectors is fundamental in many areas of science, mathematics, and engineering, as they help to describe and analyze physical quantities and systems.