VECTORS

A vector is a quantity that has both magnitude and direction. It is represented by an arrow. The length of the vector represents the magnitude and the arrow indicates the direction of the vector. Two vectors are equal if they have the same direction and magnitude (length). Blue and orange vectors have same magnitude but different direction. Blue and green vectors have same direction but different magnitude. Blue and purple vectors have same magnitude and direction so they are equal.

Vectors are often used to represent forces of varying magnitudes and angles. This figure shows 2 baseballs hit at two different angles. The length of the vector (magnitude) is the speed the baseball is hit.

P Q Initial Point Terminal Point magnitude is the length direction is this angle How can we find the magnitude if we have the initial point and the terminal point? The distance formula How can we find the direction? (Is this all looking familiar for each application? You can make a right triangle and use trig to get the angle!)

Q Terminal Point direction is this angle Although it is possible to do this for any initial and terminal points, since vectors are equal as long as the direction and magnitude are the same, it is easiest to find a vector with initial point at the origin and terminal point (x, y). If we subtract the initial point from the terminal point, we will have an equivalent vector with initial point at the origin. P Initial Point A vector whose initial point is the origin is called a position vector

To add vectors, we put the initial point of the second vector on the terminal point of the first vector. The resultant vector has an initial point at the initial point of the first vector and a terminal point at the terminal point of the second vector (see below--better shown than put in words). Initial point of V Move W over keeping the magnitude and direction the same. To add vectors, we put the initial point of the second vector on the terminal point of the first vector. The resultant vector has an initial point at the initial point of the first vector and a terminal point at the terminal point of the second vector (see below--better shown than put in words). Terminal point of w

Vectors are denoted with bold capital letters (a 1, a 2 ) This is the notation for a position vector. This means the point (a 1, a 2 ) is the terminal point and the initial point is the origin. We use vectors that are only 1 unit long to build position vectors. i is a vector 1 unit long in the x direction and j is a vector 1 unit long in the y direction. (3, 2)

The negative of a vector is just a vector going the opposite way. A number multiplied in front of a vector is called a scalar. It means to take the vector and add together that many times.

Using the vectors shown, find the following:

If we want to add vectors that are in the form ai + bj, we can just add the i components and then the j components. Let's look at this geometrically: When we want to know the magnitude of the vector (remember this is the length) we denote it Can you see from this picture how to find the length of v?

A unit vector is a vector with magnitude 1. If we want to find the unit vector having the same direction as a given vector, we find the magnitude of the vector and divide the vector by that value. If we want to find the unit vector having the same direction as w we need to divide w by 5. Let's check this to see if it really is 1 unit long.

If we know the magnitude and direction of the vector, let's see if we can express the vector in ai + bj form. As usual we can use the trig we know to find the length in the horizontal direction and in the vertical direction.

RULES FOR WORKING WITH VECTORS This says if you have a scalar (number) in front of a vector you can distribute it to each component.

This says if you want to add two vectors, you just add the corresponding components.

This says if you want to subtract two vectors, you just subtract the corresponding components.

The definition of the product of two vectors is: 1 This is called the dot product. Notice the answer is just a number NOT a vector.

Summary of Vector Rules

The dot product is useful for several things. One of the important uses is in a formula for finding the angle between two vectors that have the same initial point. u v  Technically there are two angles between these vectors, one going the "shortest" way and one going around the other way. We are talking about the smaller of the two.

Find the angle between the vectors v = 3i + 2j and w = 6i + 4j The vectors have the same direction. We say they are parallel because remember vectors can be moved around as long as you don't change magnitude or direction. What does it mean when the angle between the vectors is 0?

Determine whether the vectors v = 4i - j and w = 2i + 8j are orthogonal. The vectors v and w are orthogonal. If the angle between 2 vectors is, what would their dot product be? Since cos is 0, the dot product must be 0. Vectors u and v in this case are called orthogonal. (similar to perpendicular but refers to vectors). compute their dot product and see if it is 0 w = 2i + 8j v = 4i - j