Section 10.4 Vectors


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1 Section 10.4 Vectors A vector is represented by using a ray, or arrow, that starts at an initial point and ends at a terminal point. Your textbook will always use a bold letter to indicate a vector (such as v), but since we can't write in bold, we put an arrow over the letter that represents a vector (such as v or v ). When writing, I will use the latter notation (like the top half of a horizontal arrow) because it is the quickest to write and also because this is the notation that physicists typically use, and we learn about vectors mainly for use in future physics classes. A vector has both a magnitude (length) and direction. The arrowhead is always at the terminal point of the vector and it shows you which direction a vector has traveled. Two vectors are equal if they have the same magnitude and direction, even if they don't have the same initial and terminating points. All three vectors in the figure to the right are equal to one another. Two vectors that have the same magnitude but exactly opposite directions are denoted as v and v. When we talk about the direction of a vector we are really talking about two things: the angle the vector makes (you might think of this as its slope) and also whether it is pointing up, down, left, or right. To have "opposite directions" the slope must be the same (so the vectors must be ) but the pointing direction must be opposite. Let's look at some examples. In the top box of the figure to the left, both vectors are clearly the same length, so they have the same magnitude. Additionally, the vectors are parallel to one another, so they have the same slope. One is pointing right while the other is pointing left. So these are opposite vectors. In the second box, the vectors have the same magnitude and the same slope. One is pointing up and the other is pointing down. So these are opposite vectors also. In the last box, the vectors have the same magnitude, but they are not parallel. Because they do not have the same slope, these are not opposite vectors. ADDING VECTORS To add vectors v w, you place the initial point of w at the terminal point of v. Then the resultant sum vector v w is the vector that takes you from the initial point of v to the terminal point of w, as shown in the figure to the right. The order you add vectors in does not matter, so the vector v w is the same as the vector w v. Try this in the figure to the right. Draw w, then add v to it. Notice that the vector you get has the exact same magnitude and direction as the vector v w does. This means that vector addition is commutative Page 1
2 Vector addition is also associative, as shown in the figure to the right. Whether you add u v then w, or v w then u, or u w then v (not shown), the resultant vector will still have the same magnitude and direction. The zero vector, 0, has the property that v 0 0 v v. Also, if you add a vector, v, to its opposite vector, v, you get the zero vector: v v 0. The difference of two vectors is defined as follows: v w v w. So we never actually subtract vectors; instead, we add the opposite of the second vector. v and w are shown in the figure to the left. To find the difference v w, we add w to the terminal point of v. The resultant vector v w starts at the initial point of v and ends at the terminal point of w. MULTIPLYING VECTORS BY NUMBERS In the world of vectors, we call real numbers scalars. Scalars have only magnitude; they do not have direction. If we multiply a positive scalar () times a vector (v ), the magnitude becomes times the magnitude of the vector, but the direction is unchanged. If we multiply a negative scalar () times a vector (v ), the magnitude becomes times the magnitude of the vector and the direction changes to the opposite of v. Draw the vector 2v in the figure above. GRAPHING VECTORS Example: Given the vectors u, v, and w shown in the lefthand box below, graph the following: a) v w b) u v c) 2w d) 3v e) u 2w v u w v MAGNITUDES OF VECTORS AND THE UNIT VECTOR As we've already learned, the magnitude of a vector is its length. We will use the notation v to represent the magnitude of v. A magnitude is always positive (since there is no such thing as a negative length). A vector u that has a magnitude of 1 u 1 is called a unit vector Page 2
3 FINDING A POSITION VECTOR In order to graph the vectors in the previous example, we had to count how many units right or left and up or down a vector traveled. For instance, from its initial point to its terminal point, the vector w moved right 3 and down 2. If we assumed that the initial point was at the origin, then the terminal point would be at. But because this is a vector, we use the notation v a, b, where a and b are called the components of the vector. A vector written in this manner v a, b is called an algebraic vector. The specific case where a vector has its initial point at the origin is called a position vector. Recall that a unit vector has a magnitude of. If we say that i is the unit vector that points directly along the xaxis and j is the unit vector that points directly along the yaxis (as shown in the figure to the right), then we can rewrite v a, b as v ai bj, which indicates we move a units to the right or left and b units up or down. Label the vectors u, v, and w from the previous page in their component form, either a, b or ai bj. u v w A vector whose initial point is not at the origin can be rewritten algebraically to find an equivalent position vector by subtracting the components of the initial point from the components of the terminal point. Example: The vector v has initial point P and terminal point Q. Write v in the form ai bj (i.e. find its position vector). a) P = (3, 2); Q = (6, 5) (b) P = (1, 4); Q = (6, 2) v 6 3,5 2 9,3 v 9i 3j (c) P = (1, 1); Q = (2, 2) The magnitude of a vector v ai bj can be found using the formula v a b. Example: Find v. a) v 5i 12 j b) v i j 10.4 Page 3
4 ADDING AND SUBTRACTING VECTORS ALGEBRAICALLY Addition and subtraction of vectors is very straightforward: you add or subtract the like components together (i's with i's, j's with j's). To multiply a scalar by a vector, you multiply the scalar by each component of the vector (you can think of it as distributing). Example: If v 3i 5j and w 2i 3 j, find the following: a) 3v 2w 3 3i 5j i 3 j 9i 15j 4i 6 j 9 4 i 15 6 j 13i 21 j or 13, 21 b) 4v 3w c) v w a1, b2 i j i j i j i j d) v w FINDING A UNIT VECTOR In some applications it will be helpful to be able to find a unit vector (remember a unit vector has a magnitude of 1) that v has the same direction as a given vector. To do this, you simply divide a vector by its magnitude. So u. v Example: Find the unit vector in the same direction as v. a) v 5i 12 j b) v 2i j v Then 5 12 u i j Page 4
5 FINDING A VECTOR FROM ITS MAGNITUDE AND DIRECTION In many applications, a vector is described by its magnitude (often a speed or force) and direction (an angle, ) rather than its i and j components. In these cases it is necessary to determine the algebraic form of the vector using the formula: Example: Write the vector v in the form ai bj, given its magnitude v and the angle it makes with the positive xaxis. a) v 8, 45 2 Distribute Since cos45 sin45, we have v 8 i j v 4 2 i 4 2 j 2 b) v 3, 240 ANALYZING OBJECTS IN STATIC EQUILIBRIUM Forces can be represented by vectors, and when two forces act simultaneously on an object, their components add together to create a resultant force F1 F2. An object is said to be in static equilibrium if the object is at rest and the sum of all forces acting on the object is zero. Example: #68) A weight of 800 pounds is suspended from two cables. The left cable makes an angle of 35 and the right cable makes an angle of 50 with the beam. What are the tensions in the two cables? F pounds F 3 F 2 Assuming the weight is stationary, there are three forces creating static equilibrium: the force (tension) of each cable F1 and F2, and the force of the weight itself F 3. We need to determine the algebraic components of F1 and F2. We already know the magnitude (force) of the weight itself: it has a force of 800 pounds, being pulled straight down (due to gravity), so its force can be written as F3 800 j. We need to find the angle that each vector makes with the positive xaxis so we can plug them into the formula F F cos i sin j. Assuming the point where the weight attaches to the cables is the origin, label the angles in the figure above. 1 = 2 = 10.4 Page 5
6 Example (continued): We do not know the magnitude of the force (tension) in each cable (this is what the problem asks us to find). So we will leave them as F 1 and F 2 in the formulas. We use a calculator to find decimal approximations of cosine and sine. F F cos145i sin145 j F i j F F i F j F F cos50i sin50 j F i j F F i F j Because these three forces are in static equilibrium, their sum must equal zero. This means that the sum of each component must equal zero. F F F F F i F F 800 j We start by finding a relationship between F 1 and F 2 using the i component: i component: F F F F F F We plug this result in for F 2 when working with the j component: j component: F F F F Since F F F F F 800 F lb F lb, then F F 2 Example: Repeat the previous problem if the lefthand cable is attached to the beam at a 30 angle, the righthand cable is attached to the beam at a 55 angle, and the weight being suspended is 950 pounds Page 6
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