Chapter 7: Vectors and the Geometry of Space Section 7.1 Vectors in the Plane Written by Dr. Julia Arnold Associate Professor of Mathematics Tidewater Community College, Norfolk Campus, Norfolk, VA With Assistance from a VCCS LearningWare Grant

To represent the above quantities we use a directed line segment. In this first lesson on vectors, you will learn: Component Form of a Vector Vector Operations; Standard Unit Vectors; Applications of Vectors. What is a vector? Many quantities in geometry and physics can be characterized by a single real number: area, volume, temperature, mass and time. These are defined as scalar quantities. Quantities such as force, velocity, and acceleration involve both magnitude and direction and cannot be characterized by a single real number. To represent the above quantities we use a directed line segment.

What is a directed line segment? First let us look at a directed line segment: Q This line segment has a beginning, (the dot) and an ending (the arrow point). P We call the beginning point the “initial point” . Here we have called it P. The ending point (arrow point) is called the “terminal point” and here we have called it Q. The vector is the directed line segment and is denoted by In some text books vectors will be denoted by bold type letters such as u, v, or w.

However, we will denote vectors the same way you will denote vectors by writing them with an arrow above the letter. It doesn’t matter where a vector is positioned. All of the following vectors are considered equivalent. because they are pointing in the same direction and the line segments have the same length.

How can we show that two vectors and are equivalent? Suppose is the vector with initial point (0,0) and terminal point (6,4), and is the vector with initial point (1,2) and terminal point (7,6) . Since a directed line segment is made up of its magnitude (or length) and its direction, we will need to show that both vectors have the same magnitude and are going in the same direction. Looks verify but are not proof.

How can we show that two vectors and are equivalent? The symbol we use to denote the magnitude of a vector is what looks like double absolute value bars. Thus represents the magnitude or length of the vector . Both vectors have the same length, verified by using the distance formula. To show that the two vectors have the same direction we compute the slope of the lines. (0,0) and (6,4) Since they are also equal we (1,2) and (7,6) . Conclude the vectors are equal

What is standard position for a vector in the plane? Since all vectors of the same magnitude and direction are considered equal, we can position all vectors so that their initial point is at the origin of the Cartesian coordinate system. Thus the terminal point would represent the vector. Would be the vector whose terminal point would be (v1,v2) and initial point (0,0) The notation is referred to as the component form of v. v1 and v2 are called the components of v. If the initial point and terminal point are both (0,0) then we call this the zero vector denoted as .

Here is the formula for putting a vector in standard position: If P(p1,p2) and Q(q1,q2) represent the initial point and terminal point respectively of a vector, then the component form of the vector PQ is given by: And the length is given by:

Special Vectors If represents the vector v in standard position from P(0,0) to Q(v1,v2) and if the length of v, Then is called a unit vector. The length of a vector v may also be called the norm of v. If then v is the zero vector .

Vector Operations: Now we need to define vector addition and scalar multiplication. We will start with addition and look at the geometric interpretation. Move the first vector into standard position.

Vector Operations: Now we need to define vector addition and scalar multiplication. We will start with addition and look at the geometric interpretation.

Vector Operations: Now we need to define vector addition and scalar multiplication. We will start with addition and look at the geometric interpretation.

Vector Operations: Now we need to define vector addition and scalar multiplication. We will start with addition and look at the geometric interpretation. Move the second vector so that its initial point is at the terminal point of the first vector.

Vector Operations: Now we need to define vector addition and scalar multiplication. We will start with addition and look at the geometric interpretation. Move the second vector so that its initial point is at the terminal point of the first vector.

Vector Operations: Now we need to define vector addition and scalar multiplication. We will start with addition and look at the geometric interpretation. Move the second vector so that its initial point is at the terminal point of the first vector.

Vector Operations: Now we need to define vector addition and scalar multiplication. We will start with addition and look at the geometric interpretation. Move the second vector so that its initial point is at the terminal point of the first vector.

Vector Operations: Now we need to define vector addition and scalar multiplication. We will start with addition and look at the geometric interpretation. Move the second vector so that its initial point is at the terminal point of the first vector.

Vector Operations: Now we need to define vector addition and scalar multiplication. We will start with addition and look at the geometric interpretation. Move the second vector so that its initial point is at the terminal point of the first vector.

Vector Operations: Now we need to define vector addition and scalar multiplication. We will start with addition and look at the geometric interpretation. Move the second vector so that its initial point is at the terminal point of the first vector.

Vector Operations: Now we need to define vector addition and scalar multiplication. We will start with addition and look at the geometric interpretation. Move the second vector so that its initial point is at the terminal point of the first vector.

Now we need to define vector addition and scalar multiplication. Vector Operations: Now we need to define vector addition and scalar multiplication. The result or the resultant vector is the one with initial point the origin and the terminal point at the endpoint of vector v. Is written in standard position. See that the resultant vector can be found by adding the components of the vectors, u and v.

Vector Operations: Now we need to define vector addition and scalar multiplication. Notice, that if vector v is moved to standard position. The resultant vector becomes the diagonal of a parallelogram.

Vector Operations: Now we need to define vector addition and scalar multiplication. Notice, that if vector v is moved to standard position. The resultant vector becomes the diagonal of a parallelogram.

If and then the vector sum of u and v is Next we look at a scalar multiple of a vector, Example: Suppose we have a vector that we double. Geometrically, that would mean it would be twice as long, but the direction would stay the same. Thus only the length is affected. If then

If and then the vector sum of u and v is If and k is a scalar then Since -1 is a scalar, the negative of a vector is the same as multiplying by the scalar -1. So, Example: The negative of the vector would become Making the terminal point in the opposite direction of the original terminal point.

If and then the vector sum of u and v is If and k is a scalar then The negative of is Lastly, we examine the difference of two vectors: using the definition of the sum of two vectors and the negative of a vector.

Geometrically, what is the difference? Let u and v be the vectors below. What is u – v? vectors u and v are in standard position. Now, create the vector -v

Geometrically, what is the difference? Let u and v be the vectors below. What is u – v? vectors u and v are in standard position. Now, create the vector -v

Geometrically, what is the difference? Use the parallelogram principle to draw the sum of u - v

How do and relate to our parallelogram?

How do and relate to our parallelogram?

How do and relate to our parallelogram?

How do and relate to our parallelogram?

How do and relate to our parallelogram?

How do and relate to our parallelogram? They are both diagonals of the parallelogram.

If and then the vector sum of u and v is If and k is a scalar then The negative of is If and then the vector difference of u and v is

Vector Properties of Operations Let be vectors in the plane and let c, and d be scalars. The commutative property: The associative property: Additive Identity Property: Additive Inverse Property: Associative Property with scalars: Distributive Property: Distributive Property: Also

The length of a scalar multiple of a vector is the length of the vector times the scalar as was shown earlier and here again. If then Every non-zero vector can be made into a unit vector: Proof: First we will show that has length 1. Since is just a scalar multiple of , they are both going in the same direction.

The process of making a non-zero vector into a unit vector in the direction of is called the normalization of . Thus, to normalize the vector , multiply by the scalar . Example: Normalize the vector and show that the new vector has length 1. Multiply by Now we will show that the normalized vector has length 1.

Standard Unit Vectors The unit vectors <1,0> and <0,1> are called the standard unit vectors in the plane and are denoted by the symbols respectively. Using this notation, we can write a vector in the plane in terms of the vectors as follows: is called a linear combination of . The scalars are called the horizontal and vertical components of respectively.

Writing a vector in terms of sin and cos . Let be a unit vector in standard position that makes an angle with the x axis. Thus

Writing a vector in terms of sin and cos continued. Let be a non-zero vector in standard position that makes an angle with the x axis. Since we can make the vector v a unit vector by multiplying by the reciprocal of its length it follows that Example: Suppose vector v has length 4 and makes a 30o angle with the positive x-axis. First we use the radian measure for

Sample Problems Example 1: Find the component form of the vector v and sketch the vector in standard position with the initial point at the origin. (-1,4) (3,1)

Sample Problems Example 1: Find the component form of the vector v and sketch the vector in standard position with the initial point at the origin. (-1,4) <-4,3> (3,1)

Example 2: Given the initial point <1,5> and terminal point <-3,6>, sketch the given directed line segment and write the vector in component form and finally sketch the vector in standard position.

Example 2: Given the initial point <1,5> and terminal point <-3,6>, sketch the given directed line segment and write the vector in component form and finally sketch the vector in standard position. Solution: <-3-1,6-5>=<-4,1>

Example 3: Use the graph below to sketch

Example 3: Use the graph below to sketch First double the length of Next move into standard position. Now move into standard position Complete the parallelogram and draw the diagonal.

Example 4: Compute For

Example 4: Compute For Solution:

Example 5: Compute For

Example 5: Compute For Solution:

Example 6: For each of the following vectors, a) find a unit vector in the same direction b) write the vector in polar coordinates i.e. 1. 2. 3. 4.

Example 6: For each of the following vectors, a) find a unit vector in the same direction b) write the vector in polar coordinates 1. 2. 3. 4.

Example 7: Suppose there are two forces acting on a skydiver: gravity at 150 lbs down and air resistance at 140 lbs up and 20 lbs to the right. What is the net force acting on the skydiver? 150 20 140

Example 7: Suppose there are two forces acting on a skydiver: gravity at 150 lbs down and air resistance at 140 lbs up and 20 lbs to the right. What is the net force acting on the skydiver? 150 The net force is the sum of the three forces acting on the skydiver. Gravity would be -150j Air Resistance would be 140j The force to the right would be 20i The sum would be which would be 10 pounds down and 20 pounds to the right. 20 140 Note: The 150 lbs represents the length of the vector. A unit vector pointing in the same direction is or -j. Thus in polar coordinates the vector would be 150(-j)=-150j

Example 8: Suppose two ropes are attached to a large crate Example 8: Suppose two ropes are attached to a large crate. Suppose that rope A exerts a force of pounds on the crate and rope B exerts a force of . If the crate weighs 275 lbs., what is the net force acting on the crate? Based on your answer, which way will the crate move. A B

Example 8: Suppose two ropes are attached to a large crate Example 8: Suppose two ropes are attached to a large crate. Suppose that rope A exerts a force of pounds on the crate and rope B exerts a force of . If the crate weighs 275 lbs., what is the net force acting on the crate? Based on your answer, which way will the crate move. A B Solution: The weight of the crate combined with gravity creates a force of -275j or<0,-275>. Adding the 3 vectors we get <13, 17 > 13 lbs right and 17 lbs up

Example 9: Find the horizontal and vertical components of the vector described. A jet airplane approaches a runway at an angle of 7.5o with the horizontal, traveling at a velocity of 160 mph.

Example 9: Find the horizontal and vertical components of the vector described. A jet airplane approaches a runway at an angle of 7.5o with the horizontal, traveling at a velocity of 160 mph. 7.5o Solution: Remembering that speed is length of vector, we know that this vector is 160 miles in length. Using polar coordinates the vector is

Example 10: A woman walks due west on the deck of a ship at 3 miles per hour. The ship is moving north at a speed of 22 miles per hour. Find the speed and direction of the woman relative to the surface of the water. N W E S

Example 10: A woman walks due west on the deck of a ship at 3 miles per hour. The ship is moving north at a speed of 22 miles per hour. Find the speed and direction of the woman relative to the surface of the water. N Ship 22mph W E Woman 3 mph What angle is made by the woman relative to polar coordinates? S In unit vector terms this would be <-1,0> radians What angle is made by the ship relative to polar coordinates? radians In unit vector terms this would be <0,1> Woman vector = Ship vector= Adding the two vectors

Example 10: A woman walks due west on the deck of a ship at 3 miles per hour. The ship is moving north at a speed of 22 miles per hour. Find the speed and direction of the woman relative to the surface of the water. N Ship 22mph W E Woman 3 mph S Woman vector = Ship vector= Adding the two vectors Notice this does not answer yet the question of speed and direction. Speed is vector magnitude and direction should be in degrees with a compass direction so how do we get that?

Example 10: A woman walks due west on the deck of a ship at 3 miles per hour. The ship is moving north at a speed of 22 miles per hour. Find the speed and direction of the woman relative to the surface of the water. Ship 22mph This angle Woman 3 mph Resultant Vector Magnitude: To find direction we need an angle: This would be our reference angle in Q3 When giving directions such as NW or SE you always begin with North or South and the angle is measured from either North or South. So is not the angle we would use to give the direction. We would use its complement which is 7.77o and say the woman is walking 22.2 mph in the direction N7.770W.

Exercise 11. Given the vector with magnitude and direction following Exercise 11. Given the vector with magnitude and direction following. Write the vector in component form. Exercise 12. Given the vector in component form write the magnitude and direction of the vector with respect to N, NE, NW, S, SE, or SW direction. 3i – 4j.

Exercise 11. Given the vector with magnitude and direction following Exercise 11. Given the vector with magnitude and direction following. Write the vector in component form. Is in quadrant II with reference angle 60 degrees and from the positive x axis 120 degrees, Thus the vector is Exercise 12. Given the vector in component form write the magnitude and direction of the vector with respect to N, NE, NW, S, SE, or SW direction. 3i – 4j. Magnitude is: This vector is in Quadrant IV, (+,-) and Since 53.13 degrees would also be the reference angel between the vector and the positive x – axis, we would need to subtract from 90 degrees to find the angle between the vertical South and the vector for giving the direction of

Your Homework for this section is in Blackboard under Assignments Button. Click on Assignment 7.1