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Vectors in Physics

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Understanding scalar and vector quantities is fundamental in physics and geometry. Scalars have magnitude only, like temperature and mass, while vectors have both magnitude and direction, such as displacement and force. This text delves into graphical vector representation in planes, calculating vector magnitude and direction, vector operations like addition and subtraction, and their applications in solving geometric and physical problems.

Distinguishing Scalar and Vector Quantities

In the realm of physics, it is crucial to differentiate between scalar and vector quantities to understand the nature of various physical phenomena. Scalar quantities are defined solely by a magnitude, which is a numerical value representing size or amount, without any associated direction. Examples of scalar quantities include temperature, mass, and distance. On the other hand, vector quantities are characterized by both a magnitude and a direction. For instance, displacement, force, and acceleration are vector quantities because they describe not only how much and how large but also in which direction the quantity acts.
Three-dimensional vector space with perpendicular red, blue, and green arrows on a faint grid background, illustrating an XYZ coordinate system.

Graphical Representation of Vectors in a Plane

When representing vectors in a two-dimensional plane, we use a coordinate system, typically the Cartesian xy-plane, to specify their direction and magnitude. A vector in this plane can be depicted as an arrow pointing from one point to another, with its length proportional to the vector's magnitude. Mathematically, a two-dimensional vector can be expressed as an ordered pair (x, y) or in terms of unit vectors \(\vec{i}\) and \(\vec{j}\), where \(\vec{i}\) is a unit vector in the direction of the x-axis and \(\vec{j}\) is a unit vector in the direction of the y-axis. Thus, a vector can be written as \(x\vec{i} + y\vec{j}\), with the coordinates x and y determining the vector's horizontal and vertical components, respectively.

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00

______, force, and acceleration are examples of ______ quantities, having both magnitude and direction.

Displacement

vector

01

A 3D vector is denoted by an ordered set of values (______, ______, ______) and can be described using the unit vectors i, j, and k.

x

y

z

02

In geometry, to combine two vectors, one must align the tail of the first vector to the ______ of the second and draw a new vector from the free tail to the free ______.

head

head

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