Introduction to Vectors

What Are Vectors

Some quantities, such as velocity or force, are defined in terms of both size (also called magnitude) and direction. For example, whenever we try to describe the movement of an object, we need two pieces of information; the object's speed and the direction in which it is traveling. A quantity that has both magnitude and direction is called a vector.

Vectors Representation And Notation

A vector can be represented visually by a directed line segment (an arrow). The starting point of the line segment is called the initial point or tail of the vector, and the ending point (the tip of the arrow) is called the terminal point or head of the vector. An arrow from the initial point to the terminal point indicates the direction of the vector, and the length of the line segment represents its magnitude.

A vector is defined by its magnitude and direction, regardless of where its initial point is located. Vectors that have the same magnitude and direction are called equivalent vectors. We treat equivalent vectors as equal, even if they have different initial points. Thus, if vectors v and w are equivalent, we write v = w

There are various symbols and notations that we use to distinguish vectors from other quantities:

• Lowercase, boldfaced type, with or without an arrow on top such as v or .
• Given initial point P and terminal point Q, a vector can be represented as . The arrowhead on top is what indicates that it is not just a line, but a directed line segment.
• Given an initial point of (0,0) and terminal point (a,b), a vector may be represented as (a,b).

This last notation (a,b) has special significance. It is called the standard position. The position vector has an initial point (0,0) and a terminal point (a,b). We can change any vector into the position vector, by thinking of the change in the x-coordinates and the change in the y-coordinates. For example, if a vector has an initial point E (x₁,y₁) and a terminal point F (x₂,y₂), then its position vector can be found as:

Properties of vectors:

• A vector is a directed line segment with an initial point and a terminal point.
• Vectors are identified by magnitude, or the length of the line, and direction, represented by the arrowhead pointing toward the terminal point.
• The position vector has an initial point at (0,0) and is identified by its terminal point (a,b).

Drawing Vectors Example 1.1

Draw the vector with an initial point P(1,1) and a terminal point Q(6,4).

Solution

Since the vector goes from point P to point Q, we will name it vector .

Finding Position Vector Example 1.2

Given vector whose initial point is P(2,3) and terminal point is Q(7,5), find its position vector v.

Solution

The position vector is found by subtracting the x-coordinate of the end point from the x-coordinate of the initial point, and the y-coordinate of the end point from the y-coordinate of the initial point.

v = (7-2, 5-3) = (5, 2)

The position vector v begins at (0,0) and terminates at (5,2), as shown in the diagram below.

Vectors Magnitude and Direction

Before we can perform operations on a vector, we need to be able to find its magnitude and its direction. We can find a vector's magnitude using the Pythagorean Theorem, and we find its direction using the inverse tangent function.

For example, given a position vector v = (a,b), the magnitude can be found as:

The direction is equal to the angle (θ) formed with the x-axis, or with the y-axis, depending on the application. For a position vector, the direction can be found as:

θ = arctan(b/a)

Two vectors v and u are considered equal if they have the same magnitude and the same direction.

Vectors Magnitude and Direction Example 2.1

Find the magnitude and direction of vector v, which has an initial point P(3,2) and a terminal point Q(8,4).

Solution

Step 1) We find the position vector as:

v = (8-3, 4-2) = (5, 2)

Step 2) We use the Pythagorean Theorem to find the magnitude:

Step 3) We find the direction using the inverse tangent function:

θ = arctan(2/5) ≈ 21.8°

Addition and Subtraction of Vectors

Adding two vectors u and v produces a third vector u + v called the resultant vector. The sum of u + v can be represented visually by placing the initial point of v at the terminal point of u. This position corresponds to the notion that we move along the first vector and then, from its terminal point, we move along the second vector. The resultant vector travels directly from the beginning of u to the end of v in a straight path, as shown below:

The resultant vector is simply the sum of the x-components and the y-components of the two vectors: u + v = (u_x + v_x, u_y + v_y)

Vector subtraction is similar to vector addition. To find u - v, we simply reverse the direction of v and add it to u, which can be represented as u + (-v). The resultant vector starts at the initial point of u and ends at the terminal point of -v, as shown below.

Taking the inverse of a vector results in the same vector facing the opposite direction.

Adding and Subtracting Vectors Example 3.1

Given vectors g = (5, -2) and h (-1, 4), find:

a) g + h

b) g - h

Solution

a) To find the sum of vectors g and h, we add their x and y components as:

g + h = (5 + (-1), -2 + 4) = (4, 2)

b) To subtract vector h from g, we simply add g to the inverse of h:

g + (-h) = (5 + 1, -2 + (-4)) = (6, -6)

Scalar Multiplication of Vectors

Multiplying a vector by a scalar gives us a scaled version of the same vector i.e.(a vector that has the same direction but different magnitude / length). Scalar multiplication has no effect on the direction unless the scalar is negative, in which case the direction of the resulting vector is opposite the direction of the original vector.

When multiplying a vector by a scalar we simply multiply each component of the vector by the scalar. For example, to multiply vector u = (a, b) by scalar c, we get cu = (ca, cb)

Multiplying a Vector by a Scalar Example 4.1

Given vector v = (2,3), find 2v.

Solution

2v = (2x2, 2x3) = (4,6)

Component Form and Unit Vectors

Any Vector in two dimensions can be described as the sum of two component vectors: the horizontal component along the x-axis, and the vertical component along the y-axis. For example, the position vector v = (4,5), comes from adding the two vectors v_x = (4, 0), and v_y = (0, 5), as shown in the diagram below.

Finding Component Form of a Vector Example 5.1

Find the components of vector u, which has an initial point (2,3) and a terminal point (8,7).

Solution

We find the position vector:

u = (8 - 2, 7 - 3) = (6, 4)

The x-component is (6,0) and the y-component is (0,4).

Unit Vectors

A unit vector is a vector that has a magnitude (length) of one. Unit vectors are commonly denoted by a lowercase letter with a hat on top (). For any given vector a, we can find the unit vector, that points in the same direction as the given vector a, but has a magnitude of one, as:

That magnitude is a scalar quantity, and dividing by a scalar is the same as multiplying by the reciprocal of the scalar.

Finding the Unit Vector in the Direction of v Example 5.2

Find the unit vector of v = (5, 3).

Solution

Step 1) We find the magnitude:

Step 2) Now, we divide the x-component and y-component by |v| to get a unit vector in the same direction as v:

Unit vectors are often chosen to form the basis of a vector space, and every vector in the space may be represented as a vector sum of a scaled version of these unit vectors. For example, in a two-dimensional plane, the standard unit vectors are the horizontal unit vector i = (1,0) directed along the positive x-axis, and the vertical unit vector j = (0,1) directed along the positive y-axis.

Therefore, any vector in two dimensions can be written as the vector sum of a scaled version of vectors i and j. For example, vector u = (a, b), may be written as (ai + bj). This vector sum is called a linear combination of the vectors i and j.

The Dot Product of Two Vectors

There are two ways we can multiply a vector by another vector: the dot product, and the cross product. We will only get into the dot product here and leave the cross product, which is a bit more advanced, for another time.

The dot product is a multiplication of two vectors that results in a scalar. It tells us how much of one vector points in the direction of another. We represent the dot product by placing a little dot ⋅ between the two vectors, as in (u ⋅ v).

The dot product of two vectors v = (a,b) and u = (c,d) is the sum of the product of the horizontal components and the product of the vertical components.

v ⋅ u = ac + bd

Another way we can find the dot product of two vectors v = (a,b) and u = (c,d) is by multiplying the magnitude of vector |v| times the magnitude of vector |u|, then we multiply by the cosine of the angle between the two vectors.

v ⋅ u = |v| × |u| × cos(θ)

Remember that the result of dot product is a scalar and not a vector.

Frequently Asked Questions

Are vectors and scalars the same?

No, vectors have both magnitude (size) and direction, while scalars have magntiude only.

Can vectors be negative?

The negative of a vector is defined as another vector having the same magnitude but facing the opposite direction. The magnitude of a vector can never be negative, thus the negative sign describes only the direction.

What is position vector?

A position vector is a vector that starts at the origin point (0).