Name: vector components and direction
Uploaded: 2025-11-26T05:44:17.000Z
Duration: 1 min 30 s
Channel: JoVE

Vectors possess both magnitude and direction. Magnitude indicates strength, while direction specifies orientation in space. For example, a car's velocity of 50 km/hr east is a vector, defining its speed and direction of motion.Vectors are represented as directed line segments from an initial point to a terminal point, denoted with bold letters or an arrow on top. A half arrow on top is also commonly used; both conventions are valid.Two vectors are considered equal when they have the same magnitude, which indicates strength, and the same direction.Adding vectors involves positioning one vector's initial point at the other's terminal point to form the resultant. Alternatively, if both vectors originate from the same point, their sum is represented by the diagonal of the parallelogram they form, starting from that same point.Subtracting vectors involves adding the negative of a vector, which flips the vector&#8217;s direction. The result is shown by the diagonal of the parallelogram, running from the terminal point of the subtracted vector to the terminal point of the other vector.In the coordinate plane, vectors are represented as ordered pairs of horizontal and vertical components, typically enclosed in angle brackets.

vectors

Vectors

Vectors are mathematical entities characterized by both magnitude and direction. Unlike scalars, which are defined solely by magnitude, vectors represent quantities like displacement, velocity, and force, where direction is essential. Vectors are graphically represented as directed line segments, extending from an initial point to a terminal point, denoted with bold letters or arrows placed above the symbol. Two vectors are deemed equal if they share identical magnitudes and directions, regardless of their initial positions.Adding vectors involves placing the initial point of one vector at the terminal point of another. Alternatively, if the vectors start from the same point, their sum is the diagonal of the parallelogram formed. Subtracting vectors involves adding the negative of the second vector. Geometrically, the difference forms the opposite diagonal of the parallelogram.Multiplying a vector by a scalar affects its magnitude, stretching, or shrinking. A negative scalar reverses the vector's direction. For instance, if vector v is multiplied by -2, the resultant vector points in the opposite direction and is twice as long.Vectors are represented as ordered pairs of their horizontal and vertical components. If a vector v is represented in the plane with initial point A&#9001;x1, y1&#10217; and terminal point B&#9001;x2, y2&#10217;, then the vector v can be written as:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:90.79%;'><img style='aspect-ratio:1920/80;' src='https://cloudfront.jove.com/files/ftp_upload/40084/40084_Equation_1_editor_40084.svg' alt='40084_Equation_1.svg' /></figure>The magnitude of a vector v = &#9001;a, b&#10217; is derived using the Pythagorean theorem:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:70.56%;'><img style='aspect-ratio:1920/91;' src='https://cloudfront.jove.com/files/ftp_upload/40084/40084_Equation_2_editor_40084.svg' alt='40084_Equation_2.svg' /></figure>Vectors can also be expressed as linear combinations of unit vectors. A unit vector is a vector of length one, typically denoted as i for the horizontal component and j for the vertical component. A vector v can be written as:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:91.24%;'><img style='aspect-ratio:1920/75;' src='https://cloudfront.jove.com/files/ftp_upload/40084/40084_Equation_3_editor_40084.svg' alt='40084_Equation_3.svg' /></figure>Here, a&#8203; and b are the horizontal and vertical components, respectively. These components can also be derived from the vector's magnitude and direction angle &#952;:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:60.53%;'><img style='aspect-ratio:1920/219;' src='https://cloudfront.jove.com/files/ftp_upload/40084/40084_Equation_4_editor_40084.svg' alt='40084_Equation_4.svg' /></figure>This representation simplifies calculations involving vector addition, subtraction, and scalar multiplication, especially when dealing with vector components in the Cartesian plane.

Vectors are mathematical entities characterized by both magnitude and direction. Unlike scalars, which are defined solely by magnitude, vectors represent ...

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KEY TERMS AND CONCEPTS

vector components and direction

Vectors describe quantities with both magnitude and direction. This makes them different from scalars, which have magnitude only. Common examples include displacement, velocity, and force, where direction matters.

A vector is shown as a directed line segment from an initial point to a terminal point. It may also be written with bold letters or with an arrow above the symbol. Two vectors are equal if they have the same magnitude and direction, even when they start in different places.

Vector addition uses tip-to-tail placement. One vector is moved so its initial point begins at the terminal point of another vector. If both vectors start at the same point, their sum is the diagonal of the parallelogram they form.

Vector subtraction is done by adding the negative of the second vector. Geometrically, the difference is the opposite diagonal of the same parallelogram. Multiplying a vector by a scalar changes its length, so the vector can stretch or shrink. A negative scalar also reverses the direction. For example, multiplying vector v by -2 makes the new vector point the opposite way and makes it twice as long.

Vectors can be written as ordered pairs of horizontal and vertical components. If a vector has initial point A⟨x1, y1⟩ and terminal point B⟨x2, y2⟩, it can be expressed from those coordinates. The magnitude of v = ⟨a, b⟩ is found with the Pythagorean theorem.

Vectors can also be written with unit vectors. A unit vector has length one. The standard unit vectors are i for the horizontal direction and j for the vertical direction. In this form, v = ai + bj, where a and b are the horizontal and vertical components.

Those components can also be found from the vector’s magnitude and direction angle θ. This component form makes vector addition, subtraction, and scalar multiplication easier in the Cartesian plane. It is especially useful when working with vectors through their horizontal and vertical parts.

Vector Components and Direction

polar coordinates

Plotting Points with Polar Coordinates

Polar coordinates give a way to plot points in a plane using distance and angle. Instead of x and y values, this system uses a radial distance and an angular direction. It is especially useful for problems with circular or rotational symmetry, including waves, oscillations, and orbital paths in physics and engineering. A point in polar form is written as P(r, θ). The value r is the distance from the pole, which plays the same role as the origin in the Cartesian plane. The angle θ is measured...

graphs of polar equations

Polar Curves: Circles, Roses, and Limacons

Polar equations can be used to graph curves with radial symmetry or repeating patterns. In the polar coordinate system, a point is described by its distance from the pole, or central point, and by an angle from the polar axis, or reference direction. This format is different from rectangular coordinates and is especially useful for certain curve shapes. A circle centered at the pole appears when the radius stays constant as the angle changes. A line through the pole appears when the angle...

Vectors describe quantities with both magnitude and direction. This makes them different from scalars, which have magnitude only. Common examples include displacement, velocity, and force, where direction matters. A vector is shown as a directed line segment from an initial point to a terminal point. It may also be written with bold letters or with an arrow above the symbol. Two vectors are equal if they have the same magnitude and direction, even when they start in different places. Vector...

the dot product

Dot Product From Direction and Angle

The dot product measures how much one vector acts along another vector’s direction. Vectors are quantities with both size and direction. This calculation gives one number that shows how aligned two vectors are. One way to find the dot product is to combine the horizontal parts and the vertical parts of the two vectors. The matching parts are multiplied, and the results are added together. This form shows why the value depends on both direction and strength. The dot product becomes larger when...

key terms and concepts

Learn how vectors use magnitude, direction, components, unit vectors, and the parallelogram rule for addition, subtraction, and scalar multiplication.

Vector Components and Direction | JoVE K12

The JoVE Math Fundamentals collection is designed to build a deep and comprehensive understanding of essential mathematics. Covering the foundations of algebra, geometry, trigonometry, and analytic reasoning, this course establishes a strong base for future studies. The collection connects core mathematical concepts with real-world practice, equipping learners to solve complex problems, master problem-solving skills, and apply abstract ideas.

math fundamentals

Vector Components and Direction

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