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Definition of a Vector in Math and Geometry

Katherine Kaylegian-Starkey, Beverly Maitland-Frett
  • Author
    Katherine Kaylegian-Starkey

    Katherine has a bachelor's degree in physics, and she is pursuing a master's degree in applied physics. She currently teaches struggling STEM students at Lane Community College.

  • Instructor
    Beverly Maitland-Frett

    Beverly has taught mathematics at the high school level and has a doctorate in teaching and learning.

This lesson defines what a vector is in math and geometry. This lesson will also cover vector operations with examples. Updated: 12/14/2021

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What is a Vector in Geometry?

What is a vector in geometry? The definition of a vector in math is a line that has direction and magnitude or size. This means that to define a line as a vector in geometry, the line must have a starting point that is directed toward an endpoint. This also means that to define a line as a vector in geometry it must have a size. This is contrary to a scalar value which is a value that only has a size. Traveling at 50 mph is scalar because there is only a magnitude, but traveling 50 mph west is a vector because there is a size and a direction. In math, a vector name is designated by either a bold variable such as v or a variable with an arrow such as {eq}\vec v {/eq}. The components of a vector are denoted with angled brackets such as {eq}<x, y> {/eq}.

Figure 1: A vector example in geometry.

A vector example in geometry.

Figure 2: An example of vectors in math. Specifically, an example of vector addition.

An example of vectors in math.

What is an Example of a Vector?

What is an example of a vector, and what is an example of a scalar? An example of a scalar is the line y = 5 + x, pictured in Figure 3. It is a scalar because it has a magnitude but no direction.

Figure 3: The line y = 5 + x is scalar because there is no direction.

The line y = 5 + x is scalar.

An example of a vector is the line {eq}\vec y = 5 \hat y + 3 \hat x {/eq}, pictured in Figure 4. It is a vector because each term has a direction.

Figure 4: The vector y = 5 + 3.

The vector y = 5 + 3.

Notice that in Figure 4 the vector has a definite length, and it is drawn with an arrow to indicate direction. The notation {eq}\hat x {/eq} is read x hat, and it is used to signify that the x and y are directions, not variables. The entire expression is read the vector y has a length of 5 in the y-direction and a length of 3 in the x-direction.

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Types of Vectors in Math

Vector math is a field of mathematics that focuses on vectors, and there are certain vectors that are frequently used. The following list defines some important vector classes that are important to vector math.

  • Unit vector - A unit vector is a vector divided by its magnitude, {eq}\vec x/|x| {/eq}. The denominator means the magnitude of x. A unit vector is annotated as {eq}\hat v {/eq}.
  • Zero vector - The zero vector is a vector with a magnitude of 0.
  • Collinear vectors - These are parallel vectors.
  • Co-initial vector - These are two vectors, such as in Figure 1, that have the same starting point.
  • Co-planar vectors - Two vectors laying on the same plane.
  • Equal vectors - Vectors are said to be equal if they have the same direction and magnitude.
  • Position vector - A vector starting at the origin and directed toward a point, P. A position vector is notated using the starting point and the endpoint such as AB.
  • Opposite vectors - Two vectors of the same magnitude but opposite direction.

Operations on Vectors with Examples

Vector math uses arithmetic and calculus to do vector operations. Arithmetic vector operations are addition, subtraction, and multiplication, and each of these vector operations will be explored in detail and vector examples will be provided in the following sections.

Addition of Vectors

Algebraically, vectors are added by adding the components, the x-direction terms and the y-direction terms, together. For example, the vectors {eq}<4, 2> {/eq} and {eq}<8, 5> {/eq} are added together by adding the x-components together, 4 + 8 = 12, and the y-components together, 2 + 5 = 7. The resulting vector, therefore, is the vector {eq}<12, 7> {/eq}.

Vectors can also be added geometrically. To geometrically add vectors, first, draw the vectors to be added tip to tail. Adding vectors is commutative so it does not matter which vector is drawn first. Next, use these vectors as the legs of a triangle and draw a hypotenuse connecting the vectors. This hypotenuse is the resultant. Figure 5 follows these steps, drawing the vector {eq}<4, 2> {/eq} in orange, the vector {eq}<8, 5> {/eq} in red, and the resultant bright blue.

Figure 5: Geometric addition of vectors.

Geometric addition of vectors.

Figure 5 also draws a right triangle in dark blue, using the resultant as the hypotenuse. The leg of the triangle corresponding to the x-component of the resultant is 12 units long, and the leg of the triangle corresponding to the y-component of the resultant is 7 units long. This is exactly what was found by algebraically adding components.

Subtraction of Vectors

In addition, algebraic subtraction of vectors requires subtracting x-components from x-components and y-components from y-components. For example, the vector {eq}<3, 4> {/eq} can be subtracted from the vector {eq}<9, 6> {/eq} by subtracting the x-components, 9 - 3 = 6, and the y-components, 6 - 4 = 2. The resulting vector is the vector {eq}<6, 2> {/eq}.

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Frequently Asked Questions

What is a vector in math?

A vector in math is an object that has a magnitude and a direction. Traveling 50 mph is not a vector because it just has a magnitude, but traveling 50 mph west is a vector because it has a direction and a magnitude.

What are the examples of vector quantity?

Examples of vector quantities are quantities with a magnitude and a direction. Some examples of vectors and scalars, quantities with only a magnitude, are velocity and acceleration which are vectors but position and speed are scalars. Displacement is a vector, but weight is a scalar, and force is a vector but the temperature is a scalar.

How do you calculate a vector?

To calculate the magnitude of a vector first identify the endpoints of the vector (x_0, y_0) and (x_1, y_1). Next, subtract the x coordinates and y coordinates, (x_1 - x_0) and (y_1 - y_0), squaring the result of each. Add the squares and take the square root of the result.

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