4.8 Vectors

A vector is a directed line segment that represents both a direction and a magnitude (or length). Vectors have various applications in physics, engineering, and other fields where quantities have both magnitude and direction.

Any vector directed in two dimensions can be thought of as having an influence in two different directions. That is, it can be thought of as having two parts. Each part of a two-dimensional vector is known as a component.

A vector P_1P_2 with two components can be plotted in the xy-plane from P_1 = \left(x_1, y_1\right) to P_2 = \left(x_2, y_2\right). The vector is identified by a and b, where a = x_2 - x_1 and b = y_2 - y_1. The vector can be expressed as \left(a, b\right). A zero vector \left(0, 0\right) is the trivial case when P_1 = P_2.

For a vector represented geometrically in the plane, the components of the vector can be found using trigonometry. If the angle \theta is known, the components a and b can be calculated using the sine and cosine functions:

a=||\left(a,b\right)|| \cos \theta \\ b=||\left(a,b\right)|| \sin \theta

Examples

When we multiply a vector by a scalar (constant), the result is a new vector. The process involves multiplying the scalar by each of the components of the original vector. The new vector will be parallel to the original vector and will have the same direction if the scalar is positive, or the opposite direction if the scalar is negative. The magnitude of the new vector will be the magnitude of the original vector multiplied by the absolute value of the scalar.

Given a vector v = \left(a, b\right) and a scalar k, the product k v is a new vector kv = \left(ka, kb\right).

To find the sum of two vectors, we add their corresponding components. The resulting vector will have the sum of the original vectors' components as its components.

For example, given vectors u = \left(a_1, b_1\right) and v = \left(a_2, b_2\right), the sum u + v is a new vector w = \left(a_1 + a_2, b_1 + b_2\right).

The dot product (also called the scalar product) of two vectors is a scalar value obtained by summing the products of their corresponding components.

Given vectors u = \left(a_1, b_1\right) and v = \left(a_2, b_2\right), the dot product u \cdot v is a scalar defined as u \cdot v = a_1 \cdot a_2 + b_1 \cdot b_2.

The dot product has several important properties, such as the geometric interpretation, which relates the dot product to the magnitudes of the vectors and the angle between them. It can also be used to determine whether two vectors are perpendicular or to find the projection of one vector onto another. We will explore these properties and applications in subsequent sections.

Examples

A unit vector is a vector with a magnitude of 1. Unit vectors are used to represent the direction of a vector without considering its magnitude. In two-dimensional space \Reals^2, there are two standard unit vectors, i and j, which represent the x and y directions, respectively. The unit vector i has the components \left(1,0\right), while the unit vector j has the components \left(0,1\right).

To find a unit vector in the same direction as a given nonzero vector, we divide the vector by its magnitude. This process is called normalization. The normalization of a vector scales its components so that the resulting vector has a magnitude of 1 while preserving its direction.

For example, given a vector v = \left(a, b\right) with magnitude \left|v\right|, the unit vector u in the same direction as v is obtained by dividing v by its magnitude:

u=\dfrac{\overrightarrow{v}}{||v||}=\langle \dfrac{\overrightarrow{a}}{||v||}, \dfrac{\overrightarrow{b}}{||v||} \rangle

By expressing a vector as a linear combination of the unit vectors i and j, we can represent any vector in \Reals^2. For a given vector \left(a,b\right), its equivalent representation in terms of i and j is:

\langle a,b \rangle=a \overrightarrow{i} +b \overrightarrow{j}

This representation is useful for performing operations on vectors, such as addition, subtraction, and scalar multiplication, as well as for finding the angle between two vectors or projecting one vector onto another.

Examples

In this section, we will learn how to find the angle between two vectors and how to determine side lengths and angle measures of triangles formed by vector addition using the Law of Sines and Law of Cosines.

The dot product of two vectors can be used to find the angle between them. For two nonzero vectors v = \langle a_1, b_1\rangle and w = \langle a_2, b_2 \rangle, their dot product is defined as:

v \cdot w = a_1 \cdot a_2 + b_1 \cdot b_2

The dot product is also equal to the product of the magnitudes of the two vectors and the cosine of the angle \left( \theta \right) between them:

v \cdot w = \left|v\right| \cdot \left|w\right| \cdot \cos \left(\theta \right)

Using this relationship, we can find the angle between the two vectors by rearranging the equation:

\theta = \arccos\left(\dfrac{v \cdot w }{\left|v\right| \cdot \left|w\right|}\right)

Vector addition can create triangles, and we can use the Law of Sines and Law of Cosines to find side lengths and angle measures of these triangles. Suppose we have two vectors u and v, and their sum is w:

w=u+v

We can use the Law of Cosines to find the angle between the vectors:

\cos\left(\theta \right) = \dfrac{u \cdot v}{\left|u\right|\cdot \left|v\right|}

For side lengths, we can use the magnitudes of the vectors as the lengths of the triangle's sides. To find the other angles in the triangle, we can apply the Law of Sines:

\dfrac{\sin \left(\alpha \right)}{ \left|u\right|} = \dfrac{sin\left(\beta\right)}{\left|v\right|} = \dfrac{\sin \left(\gamma \right) }{ \left|w \right|}

Here, \alpha , \text{ } \beta, and \gamma are the angles opposite to the sides with lengths \left|u\right|, \text{ } \left|v\right|, and \left|w\right|, respectively.

By using the relationships between the angles and side lengths, we can determine the properties of triangles formed by vector addition, which is useful in various mathematical and real-world applications.

Examples