Non-collinear Vectors - Lesson I.2.2

Key Terms:

non-collinear vectors | vector resolution |

Non-collinear vectors are vectors that exist in more than one dimension (i.e. they are located along different straight lines). These vectors cannot be added algebraically; however, they can be added graphically using the tail to tip method described in the previous lesson. Sample Problem 1 Sheila left the school at noon to go jogging. She ran 3 km [S], 5 km [E], 2 km [S] and finally 4 km [W]. Determine the resultant vector. Graphical methods are useful to conceptualize a situation but because of its inherent errors it is really only useful as a first approximation for mathematical methods. If vectors form a right angle triangle (i.e. they are perpendicular vectors), Pythagorean theorem and a trigonometric ratio can be used to determine the resultant vector. Angle Explanation: will be the angle from the resultant vector to the nearest x-axis. You can determine what quadrant the angle will be in by looking at the sign of and : If is + If is + The angle will be in quadrant I. If is + If is - The angle will be in quadrant II. If is - If is + The angle will be in quadrant III. If is - If is - The angle will be in quadrant IV. Sample Problem 2 A sailboat sails 230 km [E] and then 340 km [N]. Determine its resultant vector. Solution Since these vectors are perpendicular you can use the Pythagorean theorem to determine the length of the resultant vector and a trigonometric ratio to determine the angle involved. The magnitude of the resultant vector is 410 km. Since and are positive the angle will be in quadrant I. The angle from the positive x-axis is 55.9° . Therefore using our convention for direction the angle would be expressed as [N 34.1° E]. The resultant vector would be 410 km [N 34.1° E]. Vector Resolution If vectors are not perpendicular they can be added by the vector component method. The vector is broken down into its x and y components. Step1: Write down the given information and determine the angle for each vector F, where is measured counterclockwise from the positive x-axis [principle angle] Step 2: Break each vector down into its x and y components The x component of the vector is given by Fx = F cos and the y component is given by Fy = F sin . Step 3: Add the collinear vectors algebraically. Step 4: Use the Pythagorean theorem to determine the magnitude of the resultant vector. Step 5: Use a trigonometric ratio to determine the angle of the resultant vector. Step 6: Determine the quadrant and convert this angle into north-south directions. Sample Problem 3 A football player is hoping to score a touchdown by running a complicated play. He runs 5 m [N 36° E], then changes his path to run 12 m [N 52° W] where he meets a block and is forced to run 15 m [S 73° E] before he is tackled. Determine the resultant vector. Did this player have a successful play? Step1: Write down the given information and determine the angle [principle] for each vector F. Step 2: Break each vector down into its x and y components. Step 3: Add the collinear vectors algebraically. Step 4:Use the Pythagorean theorem to determine the magnitude of the resultant vector. Step 5: Use a trigonometric ratio to determine the angle of the resultant vector. Step 6: Since both and are positive, will be in quadrant I. Convert this angle into north-south directions. The angle from north to east would be 47.8° . The resultant vector would therefore be 10.7 m [N 47.8° N]. Since the player did advance it was a successful play.