Magnitude The magnitude of a vector is represented by its length. You can multiply the magnitude of a vector by a scalar quantity to change its length. A = 10m 2A =20m 0.5A =5m

Direction All vectors must have a direction. North, south, east, west Or an angle Sometimes measured from the horizontal or vertical Sometimes measured from the positive x-axis and around to 360o C = 20m 30o C = 17m 230o B = 15m 40o

‘Adding’ Vectors Must manipulate all vectors so they are put “tip to tail” Resultant points from start to finish. C = 20m 30o B = 15m A = 10m 40o

Sketch the vector sum of A+B Must manipulate all vectors so they are put “tip to tail” Resultant points from start to finish. B = 15m 40o A = 10m

Sketch the vector sum of A+B+C Must manipulate all vectors so they are put “tip to tail” Resultant points from start to finish. C = 20m 30o B = 15m 40o A = 10m

Sketch the resultant of A-B Flip the direction of the one being subtracted Then put tip to tail and follow adding procedure B = 15m 40o A = 10m

Components of Vectors Every vector will have a… Horizontal component that points directly left or right Vertical component that points directly up or down. The components should be drawn tip to tail and lead to the same point as the original vector. The angle goes by the start. Vertical component Original vector = 25m 𝜽 Horizontal component

Calculating Vector Components In this case… Horizontal component is the adjacent side which can be calculated using 𝐴 𝑥 =𝐴 cos 𝜃 =25m cos40o =19m Vertical component is the opposite side which can be calculated using 𝐴 𝑦 =𝐴 𝑠𝑖𝑛 𝜃 = 25m sin 40o = 16m REMEMBER: horizontal is not always cosine. It depends where the angle is located! If the angle is to the vertical, sine and cosine would flip. Vertical component Original vector = 25m 𝜽=𝟒𝟎𝒐 Horizontal component

Find the components of the following vectors C = 20m 30o B = 15m 40o A = 10m 𝐴 𝑥 =𝐴 cos 𝜃 =15m cos40o =11.5m 𝑨 𝒙 =+𝟏𝟎𝒎 𝑨 𝒚 =𝟎m 𝐴 𝑦 =𝐴 𝑠𝑖𝑛 𝜃 = 15m sin 40o = 9.6m 𝐴 𝑥 =𝐴 sin 𝜃 =20m sin 30o =10.0 m 𝐴 𝑦 =𝐴 𝑐𝑜𝑠 𝜃 = 20m cos 30o = 17.3m

Find the magnitude and direction of A+B Step 1: find the components of each vector Step 2: add all of the x-components together to find the resultant’s x-component Step 3: add all of the y-components together to find the resultant’s y-component Step 4: build your actual resultant out of its components you just fund. Step 5: use Pythagorean Theorem and SohCahToa to find magnitude and direction of resultant. RESULTANT X-component: 10m+11.5m = 21.5m Y- component: 0m+9.6m = 9.6m 𝑎 2 + 𝑏 2 =𝑐 2 12.5 2 + 9.6 2 = 𝑐 2 c= 15.8m A = 10m 𝑨 𝒙 =+𝟏𝟎𝒎 𝑨 𝒚 =𝟎m 𝐴 𝑦 =𝐴 𝑠𝑖𝑛 𝜃 = 15m sin 40o = 9.6m B = 15m tan 𝜃= 𝑜𝑝𝑝 𝑎𝑑𝑗 tan 𝜃= 9.6 12.5 𝜽=𝟑𝟕.𝟓𝒐 𝐴 𝑦 =9.6m 40o 𝐴 𝑥 =𝐴 cos 𝜃 =15m cos40o =11.5m 𝐴 𝑥 =21.5𝑚