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Understanding Vector Addition: Magnitude & Direction

Learn how to add vectors algebraically by finding magnitude and direction using the Pythagorean theorem and trigonometry. Practice problems included.

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Understanding Vector Addition: Magnitude & Direction

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  1. BELLWORK 11/13/18 • Given a scale of 1 cm = 5 m/s, what kind of vectors are you adding? • Given a scale of 1 cm = 10m/s2, what kind of vectors are you adding? • Given a scale of 1 cm = 20m, what kind of vectors are you adding? • Can you add vectors of different quantities?

  2. Correct 3.2 Wksht • Take 5 minutes to correct your 3.2 Wksht using the answer key. • You will have a quiz over it tomorrow.

  3. ADDING VECTORS ALGEBRAICALLY • Finding the Magnitude and Direction of the Resultant for two vectors that form right angles to each other.

  4. Find the Magnitude Using the Pythagorean Theorem • The Pythagorean theorem is a useful method for determining the result of adding two (and only two) vectors that make a right angle to each other. • The Pythagorean theorem is a mathematical equation that allows you to find the magnitude of the resultant.

  5. Example Problem • Eric leaves the base camp and hikes 11 km, north and then hikes 11 km east. Determine Eric's resulting displacement.

  6. Use the Pythagorean Theorem

  7. YOUR TURN…..Find the magnitude of the resultant for Problem A and Problem B

  8. ANSWERS Problem A Problem B R2 = (30)2 + (40)2 R2 = 2500 R = SQRT (2500) R = 50 km • R2 = (5)2 + (10)2 • R2 = 125 • R = SQRT (125) • R = 11.2 km

  9. Use Trigonometry to find the Direction (angle) of the Resultant • The direction of a resultant vector can often be determined by use of trigonometric functions. • SOH CAH TOA is a mnemonic that helps one remember the meaning of the three common trigonometric functions - sine, cosine, and tangent functions. • Once you have found the magnitude, you can use trig to find the direction (angle) of your resultant.

  10. PRACTICE PROBLEM • 1st Step: Draw a sketch and Identify theta (angle of the resultant) • **Theta is always measured at the tail end of your resultant. • We are going to use TANGENT to solve for theta because we prefer to use numbers we did not have to derive. • Jane leaves her house and walks 10km west, and then 5 km south. Find the direction of the resultant.

  11. Be sure to put your calculators in the Degrees Mode

  12. The Calculated Angle is Not Always the Direction • The measure of an angle as determined through use of SOH CAH TOA is not always the direction of the vector. The following vector addition diagram is an example of such a situation. Observe that the angle within the triangle is determined to be 26.6º using TOA. This angle does not include the horizon to 180º • You must add 180 degrees to theta. ⍬ = 206.6º SW

  13. Your Turn!! Find theta for the resultant

  14. ANSWERS Problem A Problem B TAN ⍬= (40km / 30km) ⍬= TAN -1 53.1º Direction of R = 53.1 + 180º Direction of R = 233.1ºSW • TAN ⍬ =  (5km/10km) • ⍬ = TAN -1 (5km / 10km) • Direction of R = 26.6º + 90.0 • Direction of R = 116.6ºNW

  15. YOUR TURN………. • COMPLETE VECTOR ADDITION WKSHT

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