When 2 Vectors are at Right Angles to each other Use the Pythagorean theorem to find the magnitude of the resultant vector. (a number and an angle) R 2 = Δx 2 + Δy 2 3.2 Vector Operations (Mathematical Addition of Vectors)
The Tangent Function Use the tangent function to find the direction (angle) of the resultant vector. The inverse of the tangent function indicates the angle theta (ϴ) ϴ = tan -1 (opp) adj “SOH” = sin ϴ = opp/hyp “CAH” = cos ϴ = adj / hyp “TOA” = tan ϴ = opp / adj
Sample Problem 3A: An archaeologist climbs the Great Pyramid in Giza, Egypt. The pyramid’s height is 136 m and its width is 2.30 x 10 2 m (230 m). What is the magnitude and the direction of the displacement of the archaeologist after she has climbed from the bottom of the pyramid to the top? Given: ∆ x = 115 m (half of 230) ∆ y = 136 m R = √(115 m) 2 + (136 m) 2 R = 178 m (magnitude) R 2 = Δx 2 + Δy 2 tan θ = Δy / Δx θ = tan -1 (136 m / 115 m) θ = 49.8° (direction)
When Vectors are not Perpendicular to each other You must Resolve Vectors into 2 Components X component parallel to x axis Y component parallel to y axis Can either positive or negative values
A truck carrying a film crew must be driven at the correct velocity to enable the crew to film the underside of the plane. The plane flies at 95 km/h at an angle of 20° relative to the ground. Find the truck’s velocity. The hypotenuse (v plane ) is the resultant vector that describes the airplane’s total velocity. The adjacent leg represents the x component (v x ), which describes the airplane’s horizontal speed. The opposite leg represents the y component (v y ), which describes the airplane’s vertical speed.
We are looking for the adjacent leg Vx cos θ = adjacent / hypotenuse cos θ = v x / v plane Rearrange and solve for V x = velocity in x direction (truck’s velocity) v x = (cos θ )(v plane ) v x = (cos 20°)(95 km/h) v x = 90 km/h
Your Turn Find the components of the velocity of a helicopter traveling 95 km/h at an angle of 35° to the ground. How fast must 007 (Bond, James Bond) travel to stay beneath Dr. No’s airplane that is moving at 105 km/h at an angle of 25° to the ground? A truck drives uphill with a 15° incline. If the truck has a constant speed of 22 m/s, what are the vertical and horizontal components of the truck’s velocity.
Adding Vectors at Angles (not perpendicular) When vectors to be added are not perpendicular, they do not form sides of a right triangle.
Look at the geometry for the situation Notice, the length of the horizontal component of the resultant vector is equal to the sum of the lengths of the horizontal components of the vectors that are being added together. This is also true for the vertical component. R x1 R y1 R x2 R y2 R x1 + R x2 R y1 + R y2 Resultant Vector
Steps 1. Resolve each vector that is being added (addends) into components. 2. Add all the horizontal components together and all the vertical components together. 3. Use the Pythagorean Theorem and trig ratios to find the resultant vector.
Example: Add these 2 vectors together: 10 m/s at 0 º and 12 m/s at 25 º Find the resultant vector, R at θ 10 m/s 12 m/s 25 º R θ
Example: Find components of each vector 10 m/s 12 m/s 25 º R θ Vector 1 xy 10 cos(0)10 sin(0) Vector 2 12 cos(25)12 sin(25)
Example for Case #3 Add horizontal and vertical components 10 m/s 12 m/s 25 º R θ Vector 1 xy 10 cos(0)10 sin(0) Vector 212 cos(25)12 sin(25) 215.1
Example: Find the magnitude and angle of the resultant vector using the Pythagorean Theorem R θ R 2 = 21 2 + 5.1 2 R = 21.6 m/s θ = tan -1 (5.1/21) θ = 14 º R = 21.6 m/s at 14º 21 5.1