## Table of Contents

- What is a Vector in Geometry?
- Types of Vectors in Math
- Operations on Vectors with Examples
- The Magnitude of the Vector
- Lesson Summary

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

- What is a Vector in Geometry?
- Types of Vectors in Math
- Operations on Vectors with Examples
- The Magnitude of the Vector
- Lesson Summary

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}.

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.

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.

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.*

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.

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.

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 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.

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}.

Vectors can also be subtracted geometrically. Like for geometric vector addition, to geometrically subtract vectors they must be placed tip to tail. Unlike geometric vector addition, to geometrically subtract vectors, the opposite vector of the vector that is being subtracted must be taken before the vectors can be drawn tip to tail. After taking the opposite vector and placing the vectors tip to tail, a line can be drawn to connect the vectors, forming a triangle as in geometric vector addition. Be careful because subtraction is not commutative, and it does matter which vector is drawn first and which opposite vector is taken. Figure 6 follows these steps using the vectors {eq}<3, 4> {/eq} and {eq}<9, 6> {/eq}, with the vector {eq}<9, 6> {/eq} drawn in red, the vector{eq}<-3, -4> {/eq} drawn in orange, and the resultant drawn in light blue.

Figure 6 also draws a right triangle in dark blue, using the resultant as its hypotenuse. As expected from the algebraic subtraction, the leg of the triangle corresponding to the x-component of the resultant is 6 units and the leg of the triangle corresponding to the y-component of the resultant is 2 units.

Vectors can be multiplied by scalars, and the result is a scaled vector. To multiply a vector by a scalar, distribute the scalar to each component of the vector and multiply.

For example, the vector {eq}\vec f = <4, -9> {/eq} can be multiplied by the scalar 3,

{eq}3 \vec f = <4*3, -9*3> {/eq}

and the result is a vector in the same direction, just three times bigger,

{eq}3 \vec f = <12, -27> {/eq}.

The magnitude of a vector refers to the size of the vector, and it is written as {eq}|\vec v| {/eq}. As seen in Figure 5 and Figure 6, it is possible to think of a vector as the hypotenuse of a right triangle where the legs of the triangle are the x and y-components of the vector. Because of this vector decomposition, the magnitude of a vector can be found using the Pythagorean theorem. For example, given the vector {eq}\vec c = 6 \hat x - 8 \hat y {/eq}, the magnitude can be found by calculating

{eq}\sqrt(6^2 + (-8)^2) {/eq}

= {eq}\sqrt(36 + 64) {/eq}

= {eq}\sqrt(100) {/eq}

= {eq}10 {/eq}

In geometry, a **vector** is a line that has a **magnitude** and a direction. This is as opposed to a** scalar **which is a value that has a size but no direction. Vectors can be added, subtracted, and multiplied. Vectors can be added and subtracted algebraically by combining like components, or vectors can be added and subtracted geometrically by placing them tip to tail and drawing a hypotenuse connecting them.

A whole field of mathematics is devoted to vector calculations, and there are many different types of vectors that are commonly used. A unit vector is a special vector that divides a vector by its magnitude, and the magnitude of a vector can be found by first decomposing it into its components and then using the Pythagorean theorem.

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Frequently Asked Questions

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.

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.

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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