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Vector Operations

Vector algebra focuses on operations such as addition, subtraction, and multiplication on physical quantities characterized by both magnitude and direction. These quantities are known as vectors.

The peculiarity of vector quantities is that they encapsulate more than just a value - they also incorporate direction and orientation. As a result, you cannot manipulate them as you would regular numbers using conventional algebraic rules.

Dealing with vectors involves a unique strand of mathematics called vector algebra, setting out the guidelines for executing operations between vectors.

Operations in Vector Algebra

Two key operations can be performed on vectors: addition (or subtraction) and multiplication.

  • Vector Addition
    The process of adding two vectors is akin to merging their respective directions. Imagine walking 10 steps north from Point A to Point B (constituting one vector), followed by 5 steps east from B to C.
    example of two vectors
    The addition of these vectors gives the aggregate distance traveled, that is, the vector from A to C, which manifests as a diagonal path in this case. Hence, the net outcome of the addition of two vectors is another vector.
    vector addition
    From a mathematical perspective, the addition of vectors is achieved by summing the corresponding components of the vectors. For instance, given two vectors $$ \vec{a} = \begin{pmatrix} a_1 \\ a_2 \end{pmatrix} $$ $$ \vec{b} = \begin{pmatrix} b_1 \\ b_2 \end{pmatrix} $$ their resultant vector S is the sum of their respective components $$ \vec{a} + \vec{b} = \begin{pmatrix} a_1 \\ a_2 \end{pmatrix} + \begin{pmatrix} b_1 \\ b_2 \end{pmatrix} = \begin{pmatrix} a_1 + b_1 \\ a_2 + b_2 \end{pmatrix} $$
  • Vector Multiplication
    Within vector algebra, there exist two fundamental types of vector products: the dot product and the cross product.
    • Dot Product (Scalar or Inner Product)
      When two vectors are multiplied using the dot product, the outcome is a scalar quantity, which is just a number, not another vector. The dot product is computed by multiplying the matching components of the vectors and then adding the results. For instance, given two vectors $$ \vec{a} = \begin{pmatrix} a_1 \\ a_2 \end{pmatrix} $$ $$ \vec{b} = \begin{pmatrix} b_1 \\ b_2 \end{pmatrix} $$ their dot product P is the sum of the products of their corresponding components: $$ P = a_1 \cdot b_1 + a_2 \cdot b_2 $$ This product relates to the angle between the vectors. If the vectors are orthogonal, with the angle between them being 90 degrees, the dot product is zero.
    • Cross Product
      A cross product operation between two vectors results in a new vector. This resultant vector from the cross product has a magnitude (length) proportional to the product of the lengths of the vectors and the sine of the angle between them. Its direction is perpendicular to the plane containing the two original vectors, which is determined by the right-hand rule. The cross product is primarily relevant to vectors in three-dimensional space.

Bear in mind, operations involving vectors do not always abide by the same rules as operations with scalar numbers. For example, the product of two vectors is not commutative; changing the order of vectors in a cross product reverses the sign of the result.




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