|Module 24 - Vectors|
|Introduction | Lesson 1 | Lesson 2 | Lesson 3 | Self-Test|
|Lesson 24.1: Vector Arithmetic|
In this lesson you will learn to define vectors on the TI-89 and to perform three types of vector multiplication. Unit vectors will be discussed and two formats used to denote vectors will be identified.
Quantities that have both magnitude and direction are called vectors and are often represented by directed line segments, as illustrated below.
The vector shown has an initial point at O and a terminal point at P.
Representing Vectors using Brackets
Vectors can be represented on the TI-89 by giving the coordinates of the tip of the arrow. For example, a vector that goes from the origin to the point (3, 2) is represented on the TI-89 with the notation [3, 2]. Note the use of brackets instead of parentheses to denote that the quantity is a vector.
Defining Unit Vectors i and j
The vector i is one unit long and points along the positive x-axis and the vector j is one unit long and points along the positive y-axis. Because the vectors i and j are each one unit long they are called unit vectors. Both i and j are shown below along with the vector [3, 2].
Representing Vectors using i and j
Another notation uses the unit vectors i and j to represent a vector. The vector [3, 2] can also be written as
a = 3i + 2j
Notice that a, i, and j are written in bold to signify that they are vectors.
Finding the Length of a Vector
The length or magnitude of any vector a = [x, y] is
The length of a = [3, 2] is units.
There are three types of multiplication that involve vectors. Two types produce a vector and the remaining type produces a real number. Each type of multiplication is discussed below.
Scalar Multiplication of Vectors
Letting c represent a
The coordinates of ca are found by multiplying each coordinate of a by c.
ca = c[a1, a2] = [ca1, ca2]
Using the unitV Command
The vector that points in the same direction as a and has a magnitude of one can be found with the unitV command.
Determine a unit vector that points in the same direction as a = [3, 2].
The menu item "1:unitV(" should be highlighted.
Each component of a has been multiplied by the reciprocal of the magnitude of a to create the unit vector that points in the same direction as a. Note that the fractions have been
Finding Dot Products of Vectors
The second type of multiplication is called a dot product. The dot product of the two vectors [a1, a2] and [b1, b2] is defined to be a1 · b1 + a2 · b2.
Compute the dot product a · b.
The dot product a · b is 4.
Notice that the result of the dot product of two vectors is a real number, not a vector. The dot product is the same as the product of the magnitude of a, the magnitude of b and the cosine of the angle between a and b.
Dot products are widely used in physics. For example, they are used to calculate the work done by a force acting on an object.
Projecting One Vector onto Another Vector
A projection can be thought of as the shadow of one vector on another. When the two vectors have the same initial point, the projection of b onto a is parallel to a and has the length of the shadow of b. The diagram below illustrates the projection of b onto a, written as projab and shown as the darker vector.
Projections and Dot Products
The magnitude of the projection of b onto a, |projab|, is also called the component of b along a, and it is equal to | b | cos . Note that the component of b along a is equal to a · b / |a|.
Finding the Formula for Dot Products on your Calculator
The formula for finding the dot product of two vectors [a1, a2] and [b1, b2] can be derived on the TI-89.
24.1.1 Write the formula for finding the dot product of two vectors. Click here for the answer.
Defining Cross Products
The third type of multiplication is called a cross product and it is used in geometry and many situations in physics and engineering. The cross product a x b of two three-dimensional vectors is a vector that is perpendicular to both a and b. If is the angle between a and b, then the length of a x b is given by
The cross product is only defined for 3-dimensional vectors, but the TI-89 computes the cross product of 2-dimensional vectors by treating them as 3-dimensional vectors with 0 as the third component.
Finding Cross Products
A cross product can be calculated with the crossP command, which is found in the Math Matrix Vector ops menu.
Find the cross product a x b of the previously defined vectors a = [3, 2] and b = [-2, 5].
[3, 2, 0] x [-2, 5, 0] = [0, 0, 19] = 0i + 0j + 19k
The result of a cross product is a vector that has three components, which means that it is a three-dimensional vector. The unit vector k points along the positive z-axis. The vector 0i + 0j + 19k points upward along the positive z-axis and has a length of 19.
The cross product of two vectors a and b is always perpendicular to each of the two vectors. It has a magnitude that is equal to the magnitude of a multiplied by the magnitude of the component of b that is perpendicular to a.
24.1.2 Find the cross product of the three-dimensional vectors a = 2i + j - 4k and b = -i 2j + k. Describe the relationship between a, b, and their cross product. Click here to check your answer.
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