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Vector Mathematics Physics 1 Physical Quantities A scalar quantity is expressed in terms of magnitude (amount) only. Common examples include time, mass, volume, and temperature. Physical Quantities A vector quantity is expressed in terms of both magnitude and direction. Common examples include velocity, weight (force), and acceleration. Representing Vectors Vector quantities can be graphically represented using arrows. – magnitude = length of the arrow – direction = arrowhead Vectors All vectors have a head and a tail. Vector Addition Vector quantities are added graphically by placing them headto-tail. Head-to-Tail Method 1. 2. Draw the first COMPONENT vector with the proper length and orientation. Draw the second COMPONENT vector with the proper length and orientation starting from the head of the first component vector. Head-to-Tail Method 3. 4. The RESULTANT (sum) vector is drawn starting at the tail of the first component vector and terminating at the head of the second component vector. Measure the length and orientation of the resultant vector. To add vectors, move tail to head and then draw resultant from original start to final point. East Resultant is (sqrt(2)) 45◦ south of East South Vector addition is ‘commutative’ (can add vectors in either order) East Resultant is (sqrt(2)) 45◦ south of East South Vector addition is ‘commutative’ (can add vectors in either order) East South Resultant is (sqrt(2)) 45◦ south of East South East Co-linear vectors make a longer (or shorter) vector Resultant is 3 magnitude South Co-linear vectors make a longer (or shorter) vector Resultant is 3 magnitude South North Can add multiple vectors. Just draw ‘head to tail’ for each vector East North East Resultant is 2 magnitude 45◦ North of East North Adding vectors is commutative. Resultant is 2 2 magnitude 45◦ North of East East East North East South North East North Equal but opposite vectors cancel each other out West East Resultant=0 Resultant is 0. South Vector Addition – same direction A+B=R A B B A R=A+B Vector Addition • Example: What is the resultant vector of an object if it moved 5 m east, 5 m south, 5 m west and 5 m north? Vector Addition – Opposite direction (Vector Subtraction) . A + (-B) = R A B -B A A + (-B) = R -B Vectors • The sum of two or more vectors is called the resultant. Practice Vector Simulator http://phet.colorado.edu/sims/vector-addition/vector-addition_en.html Polar Vectors Every vector has a magnitude and direction magnitude angle dire ction Right Triangles SOH CAH TOA Vector Resolution Every vector quantity can be resolved into perpendicular components. Rectilinear (component) form of vector: xy Vector Resolution Vector A has been resolved into two perpendicular components, Ax (horizontal component) and Ay (vertical component). A Ay Ax Vector Resolution If these two components were added together, the resultant would be equal to vector A. A Ay Ax Vector Resolution When resolving a vector graphically, first construct the horizontal component (Ax). Then construct the vertical component (Ay). Using right triangle trigonometry, expressions for determining the magnitude of each component can be derived. Vector Resolution Horizontal A Component (Ax) Ay Ax Ax cos A Ax A cos Vector Resolution Vertical A Component (Ay) Ay Ax sin Ay A sin Ay A Drawing Directions EX: 30° S of W – – Start at west axis and move south 30 ° Degree is the angle between south and west N W E S Vector Resolution Use the sign conventions for the x-y coordinate system to determine the direction of each component. N (-,+) (+,+) E W (-,-) (+,-) S Component Method 1. 2. 3. Resolve all vectors into horizontal and vertical components. Find the sum of all horizontal components. Express as SX. Find the sum of all vertical components. Express as SY. Component Method 4. 5. 6. Construct a vector diagram using the component sums. The resultant of this sum is vector A + B. Find the magnitude of the resultant vector A + B using the Pythagorean Theorem. Find the direction of the resultant vector A + B using the tangent of an angle . Adding “Oblique” Vectors Head to tail method works, but makes it very difficult to ‘understand’ the resultant vector Adding “Oblique” Vectors Break each vector into horizontal and vertical components. Add co-linear vectors Add resultant horizontal and vertical components Adding “Oblique” Vectors Break each vector into horizontal and vertical components. Adding “Oblique” Vectors Break each vector into horizontal and vertical components. Add co-linear vectors Adding “Oblique” Vectors Break each vector into horizontal and vertical components. Add co-linear vectors Add resultant horizontal and vertical components Adding “Oblique” Vectors Break each vector into horizontal and vertical components. Add co-linear vectors Add resultant horizontal and vertical components Using Calculator For Vectors Can use the “Angle” button on TI-84 calculator to do vector mathematics Using Calculator for Vectors