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Many quantities in geometry and physics, such as area, volume, temperature, mass, and time, can be characterized as a single real number scaled to appropriate units of measure. These are called scalar quantities, and the real number associated with each is called a scalar.
Other quantities, such as force, velocity, and acceleration, involve both magnitude and direction and cannot be characterized completely by a single real number. A directed line segmentis used to represent such a quantity. A directed line segment has an initial point and terminal point for example PQ would have initial point P and terminal point Q, and its length or (magnitude) is denoted as ||PQ|| or sometimes as |PQ|. The set of all directed line segments that are equivalent to a given directed line segment are vectors in the plane. Vectors are denoted in typesets as boldface letters such as u, v, and w. Written by hand they are usually denoted as
Scalars • A scalar quantity is a quantity that has magnitude only and has no direction in space • Examples of Scalar Quantities: • Length • Area • Volume • Time • Mass
Vectors • A vector quantity is a quantity that has both magnitude and a direction in space • Examples of Vector Quantities: • Displacement • Velocity (speed in a given direction) • Acceleration (rate of which something speeds up or slows down in a direction) • Force (gravity for example, or pushing an object with a force of 10 Newtons (N)).
Because these two vectors are the same length but are in opposite directions we have u and –u. Vector Diagrams • Vector diagrams are shown using an arrow • The length of the arrow represents its magnitude • The direction of the arrow shows its direction u v -u Because these two vectors (u and v) are going in the same direction and have the same magnitude (length) then the vectors are the same u=v.
Resultant of Two Vectors • The resultant is the sum or the combined effect of two vector quantities Vectors in the same direction: 6 N 4 N = 10 N 6 m = 10 m 4 m Vectors in opposite directions: 6 m s-1 10 m s-1 = 4 m s-1 6 N 10 N = 4 N
The sum of two vectors can be represented geometrically by positioning the vectors (without changing their magnitudes or directions) so that the initial point of one coincides with the terminal point of the other. The vector u+v is called the resultant vector. u v u+v
If you have a vector v and then –v is a vector of the same magnitude, however it is directed in the opposite direction. This is a property what we will use to subtract two vectors such as u-v, we will change to addition of u + (-v) so you need to locate vector v, change its direction and then move it to tip-tail. -v u-v u v Animation done by Taylor Cox (the guy)
The Parallelogram Law • When two vectors are joined tail to tail • Complete the parallelogram • The resultant is found by drawing the diagonal The Triangle Law • When two vectors are joined head to tail • Draw the resultant vector by completing the triangle
Adding Vectors “Tip to Tail” method • Suppose that we have the following two vectors Graphically represent u+v. v v u u u+v Resultant vector
Graphically represent 2u+3v 3v 2u v u
Represent v-u • Is that the same thing as v+(-u)? v u v-u -u
Vector Algebra -component form -vector addition -vector subtraction -scalar multiplication
Component form of a vector is essentially working backwards to identify the two perpendicular vectors that would add (tip to tail) to provide you with the current vector. These two perpendicular vectors are essentially the x-direction and y-direction distances. These values are referred to as the COMPONENTS of your particular vector. The x component of v is 3 The y component of v is 2 Thus we write v = (3,2) or This is referred to as COMPONENT FORM of v 2 3
You can find the component form of a vector by subtracting the ordered pairs of tip and tail. General Example (x2,y2) (4,3) y2-y1 y2-y1 = 3-1 =2 (x1,y1) (1,1) x2-x1 x2-x1 =4-1=3 If we needed to find the length of v we would use the Pythagorean Theorem, and then the Pythagorean Theorem would lead us directly to the distance formula. Because we are talking about length it will be positive so we use notation that implies absolute value. This is the equation for the size or MAGNITUDE of the vector.
Vector addition (algebraically) is quite simple if your vectors are both expressed in component form. • Suppose we have two vectors v and w. In component form v=(a,b) and w=(c,d) then vector addition states that v+w=(a,b)+(c,d), which means v+w = (a+c,b+d)
Example • If v=(1,2) and w=(5,1) find v+w. • v+w = (1,2)+(5,1) = (1+5, 2+1) = (6,3) • v+w = (6,3). • Think of it graphically/geometrically. The vectors can start anywhere in the Cartesian Plane, lets assume that one starts at the origin, and the other starts at the tip of the first (you can assume that they both start at the origin and then use parallelogram technique to verify your answer) w Resultant vector v+w is purple Identify its components. Are they (6,3)? 1 5 v 2 1
Vector Subtraction • Vector subtraction is the same concept if v=(a,b) and w=(c,d) then v-w=(a-c, b-d) Remember geometrically vector subtraction is the same as addition however you add the opposite direction of the vector. If I want to subtract v-w, geometrically we would take v + (-w). And the (-w) would be the same exact vector as w but going in the opposite direction. v w
Scalar Multiplication • Refers to the idea that we can take any vector in component form and multiply it by any constant that we want to. In light of doing so, the components are both changed by the same scalar. • If v=(3,2) and we want 3v then we multiply both components by 3, thus 3v=(9,6) • The constant 3 is our SCALAR
Properties of Vectors • u+v= v+u (commutative) • u + (v + w) = (u + v) + w (associative) • u + 0 = u (identity of addition) • u + (-u) = 0 (subtraction)
Find the magnitude of the resultant vector 8 lb 39o 14 lb
10.3 Algebraic Interpretation of Vectors • A vector with its initial point at the origin is called a position vector. • A position vector u with endpoint (a,b) is written as u = a, b, where a is called the horizontal component and b is called the vertical component of u.
10.3 Finding Horizontal and Vertical Components Horizontal and Vertical Components The horizontal and vertical components, respectively, of a vector u having magnitude |u| and direction angle are given by a = |u| cos and b = |u| sin . That is, u = a, b = |u| cos , |u| sin . .
Resolving a Vector Into Perpendicular Components • When resolving a vector into components we are doing the opposite to finding the resultant • We usually resolve a vector into components that are perpendicular to each other • Here a vector v is resolved into an x component and a y component v y x
Calculating the Magnitude of the Perpendicular Components • If a vector of magnitude v and makes an angle θ with the horizontal then the magnitude of the components are: • x = v Cosθ • y = v Sinθ v y=v Sinθ y θ x=v Cos θ x • Proof:
10.3 Finding Horizontal and Vertical Components Example From the figure, the horizontal component is a = 25.0 cos 41.7° 18.7. The vertical component is b = 25.0 sin 41.7° 16.6.
Problem: Calculating the magnitude of perpendicular components 2002 HL Sample Paper Section B Q5 (a) A force of 15 N acts on a box as shown. What is the horizontal component of the force? Solution: 12.99 N 15 N Vertical Component 60º Horizontal Component 7.5 N
Problem: Resultant of 2 Vectors Two forces are applied to a body, as shown. What is the magnitude and direction of the resultant force acting on the body? A forces of 12N due East and 5N due South Solution: • Complete the parallelogram (rectangle) • The diagonal of the parallelogram ac represents the resultant force • The magnitude of the resultant is found using Pythagorean Theorem on the triangle abc 12 N a d θ 13 N 5 N 5 b c 12
Problem: Resultant of 3 Vectors Find the magnitude (correct to two decimal places) and direction of the resultant of the three forces shown below. Solution: • Find the resultant of the two 5 N forces first (do right angles first) 5 d c 7.07 N • Now find the resultant of the 10 N and 7.07 N forces 5 5 N 90º • The 2 forces are in a straight line (45º + 135º = 180º) and in opposite directions 45º θ a 2.93 N b 5 N 135º • So, Resultant = 10 N – 7.07 N = 2.93 N in the direction of the 10 N force 10 N
Practical Applications • Here we see a table being pulled by a force of 50 N at a 30º angle to the horizontal • When resolved we see that this is the same as pulling the table up with a force of 25 N and pulling it horizontally with a force of 43.3 N 50 N y=25 N 30º x=43.3 N • We can see that it would be more efficient to pull the table with a diagonal force of 50 N
Inclined Planes • A person in a wheelchair is moving up a ramp at constant speed. Their total weight is 900 N. The ramp makes an angle of 10º with the horizontal. Calculate the force required to keep the wheelchair moving at constant speed up the ramp. (You may ignore the effects of friction). Solution: If the wheelchair is moving at constant speed (no acceleration), then the force that moves it up the ramp must be the same as the component of it’s weight parallel to the ramp. Complete the parallelogram. Component of weight parallel to ramp: 156.28 N 10º 80º 10º Component of weight perpendicular to ramp: 886.33 N 900 N
10.3 Finding the Magnitude of a Resultant Example Two forces of 15 and 22 newtons act on a point in the plane. If the angle between the forces is 100°, find the magnitude of the resultant force. Solution From the figure, the angles of the parallelogram adjacent to angle P each measure 80º, since they are supplementary to angle P. The resultant force divides the parallelogram into two triangles. Use the law of cosines on either triangle. |v|2 = 152 + 222 –2(15)(22) cos 80º |v| 24 newtons
10.3 The Unit Vector • A unit vector is a vector that has magnitude 1. • Two very useful unit vectors are defined as i = 1, 0 and j = 0, 1. i, j Forms for Unit Vectors If v = a, b, then v = ai + bj.
10.3 Dot Product Dot Product The dot product of two vectors u = a, b and v = c, d is denoted u · v, read “u dot v,” and given by u · v = ac + bd. Example Find the dot product 2, 3 · 4, –1. Solution 2, 3 · 4, –1 = 2(4) + 3(–1) = 8 – 3 = 5
10.3 Applying Vectors to a Navigation Problem Example A plane with an airspeed of 192 mph is headed on a bearing of 121º. A north wind is blowing (from north to south) at 15.9 mph. Find the groundspeed and the actual bearing of the plane. Solution Let |x| begroundspeed. We must find angle . Angle AOC = 121º. Find |x| using the law of cosines .
10.3 Finding a Required Force Example Find the force required to pull a wagon weighing 50 lbs up a ramp inclined at 20º to the horizontal. (Assume no friction.) Solution The vertical 50 lb force BA represents the force of gravity. BA is the sum of the vectors BC and –AC. Vector BC represents the force with which the weight pushes against the ramp. Vector BF represents the force required to pull the weight up the ramp. Since BF and AC are equal, | AC | gives the magnitude of the required force.
