Addition, subtraction and scalar multiplication, computed algebraically and geometrically
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The equation of a vector in two dimensions can be written as a sum of its vertical and horizontal components, denoted i and j respectively. These are vectors that start from the origin and point right (i) and up (j), each with a length of 1. We call these base vectors.
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A vector in two dimensions can be uniquely described by giving its vertical and horizontal components, meaning the number of units in each direction required to get from the "tail" to the "head" of the arrow. These components can be represented using a column vector.
The column vector of a vector v whose head is located a horizontal and b vertical units from its tail is
The top number represents the horizontal component and the bottom number represents the vertical component.
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A vector is a mathematical object that has both magnitude (size) and direction in space. A three-dimensional vector is typically described by its components along the x, y, and z axes.
In 3 dimensions, a vector is expressed as
where i, j, and k are called the base vectors:
To add two vectors, add their corresponding components:
To subtract two vectors, subtract their corresponding components:
Addition corresponds to placing vectors head-to-tail, and subtraction corresponds to the displacement between their tips.
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A position vector describes the position of a point relative to a fixed origin. For example, the position vector of a point P describes the location of P relative to an origin O:
A displacement vector describes how to move from one point to another. For instance, the displacement vector from point A to point B is given by subtracting position vectors:
Thus, position vectors identify points relative to an origin, while displacement vectors represent movement or translation between points. Expressing AB as OB−OA is sometimes referred to as giving the relative position of B from A.
The zero vector 0 is a special vector without size or a defined direction, represented by
The negative of a vector reverses the vector's direction while maintaining its size. If v=⎝⎛xyz⎠⎞, then the negative is
pointing in exactly the opposite direction.
Scaling a vector means multiplying it by a number k, which changes its length without affecting its direction (unless k<0, which reverses direction). If
then scaling by k gives
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Two vectors are parallel if one can be written as a scalar multiple of the other, i.e., if u=kv for some scalar k. Parallel vectors have identical or exactly opposite directions.
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The magnitude (or length) of a vector v=⎝⎛v1v2v3⎠⎞ is calculated as
Geometrically, this represents the distance from the vector's starting point to its endpoint in 3D space. For example, the magnitude of the displacement vector AB equals the straight-line distance between points A and B.
When a vector u has a magnitude of 1, we say u is a unit vector.
We often express other vectors v with magnitude ∣v∣=1 as scalar multiples of the unit vector pointing in the same direction,
where k=∣v∣.
The unit vector in the same direction as a non-unit vector v is written as
Rescaling to form a vector with a magnitude of one is referred to as normalizing a vector.