Vectors, Addition and Scaling

What is a Vector?

A vector is a mathematical object that represents a vector in space. Let us consider the simple example of a Cartesian coordinate system (or Point). Suppose that the direction of a vector is -60, 0, and it is located along the x-axis and has length 1. Now draw a triangle with the x-axis as the hypotenuse and the angle as 90 degrees. It is a straight line since is not perpendicular to the x-axis and there is no negative angle. Assume, for simplicity, that the x-axis and the y -axis are equal to the x and y axes. Then the triangle above has dimensions x, y, and z, and each point on the x-axis is labeled x, and each point on the y -axis is labeled y. So the vector is of type (x, y, z). So a vector is simply a vector in a Cartesian coordinate system.

Vector Addition

The addition of two or more vectors is one of the basic operations in linear algebra. So how does one get started? How do we start creating vectors? Let’s start with basic addition. The addition of two or more vectors has many characteristics. It’s associative, commutative, and distributive. The more we understand scalar multiplication, the more we can use vectors. If a vector is zero-based, the two vectors will have zero multiplied together. If a vector is nonzero-based, the two vectors will have one multiplied together. A vector is either a scalar or a vector. Calculating the Sum of two or more Vector Additions There are a few ways to calculate the sum of vector addition.

Scalar and Vector Subtraction

Scalar subtraction is the operation of removing a single scalar value from a vector. For instance, the vector from (length 1) to (length 2) is invertible (there is a scalar value 1 for the pair), so the scalar value can be removed. If the scalar in the vector was removed, the result would be vector 1: Vectors in Vector Algebra Vectors can be in one of three basic cases: Point, Scalar, and Vector. The first and most fundamental case is Vector. All basic vectors are vectors (that is, they contain a scalar value that is zero for their whole range of values). For instance, a vector from (length 1) to (length 2) is vector (length 1) – (length 2). A vector can also be in the other two basic cases: Point or Scalar.

Scaling Vectors

Scaling a vector is the operation of rotating a vector about its horizontal axis (hence, the term rotational scaling) by some angle. To scale the length of a vector, the rotation angle needs to be in the range from −90° to 90°. It follows from the theorem of Pythagoras that, for the real numbers, (See "Are there other scaling rules for the real numbers?") In the picture, we see that scale the length of the vector by 90° about the horizontal axis. In other words, we rotate the vector about the vertical axis by 180°. In the picture, we see that scale the length of the vector by 90° about the horizontal axis. In other words, we rotate the vector about the vertical axis by 180°. Scalar addition is the operation of adding scalars together into a scalar product.

Conclusion

Vector addition and scalar multiplication are both quite powerful operations and require little boilerplate for operators. If you have any questions about vector addition or scalar multiplication, please leave a comment and we will do our best to answer any question that you may have!