 A scalar is a quantity that can be completely specified by its magnitude with appropriate units. A scalar has magnitude but no direction. Examples: speed, volume, the number of pages in a book  A vector is a physical quantity that has both direction and magnitude. Examples: displacement, velocity, acceleration

VECTORS Vectors can be represented in bold type with an arrow above the symbol. Vectors are represented by drawing arrows: > The length and direction of a vector should be drawn to a reasonable scale size to show its magnitude. 10km > 20km > When vectors are added together they should be drawn head to tail to determine the resultant or sum vector. 10km + 20km | > >|

VECTORS CONTINUED By measuring a scale version, the resultant of the two vectors can be determined. In subtracting vectors the vector being subtracted needs to be turned completely around before the resultant can be determined. 10km - 20km | OR 20km - 10km | |

oThe resultant vector can be found mathematically in right triangles using the Pythagorean Theorem: C² = A² + B² OR in other triangles using the Law of Cosines: C² = A² + B² - 2AB cos θ oA coordinate system has an origin and two perpendicular axes. oA single vector can be broken down into two or more component to find its x and y coordinates. Ax = A cos θ Ay = A sin θ

By resolving vectors you will also form right triangles. A² = Ax² + Ay² Ax and Ay are known as the components of a vector. If more than one vector is resolved the components of the resultant vector can be found simply by adding the components of all the vectors. Rx = Ax + Bx + Cx... Ry = Ay + By + Cy...

Objects that move in two or more dimensions have the same kinematic relationships of position, displacement, velocity, and acceleration. These vector equations can be split into components along the x-axis and y-axis. Motion in two dimensions is called projectile motion. v² = v²x + v²y

Projectile Motion Continued The vertical component of velocity is affected by the acceleration of gravity and is described by the equations for an object in free fall. v yf = v yi + gtv² yf = v² yi + 2gy Y = v yi t + 1/2 gt²v y = v SIN θ

If air resistance is negligible, then the horizontal component of velocity does not change. v xi = v xf X = v xi t v x = v COS θ Range = v xi t = v²/g SIN 2θ Projectile Motion Continued

The relative velocity of an object depends on its frame of reference. A different frame of reference implies a different velocity if the two frames are in relative motion. Problems concerning relative velocity involve the addition and subtraction of vectors. Relative velocity problems also arise when an object is moving in a medium that is itself moving, such as boats in currents or planes in wind.