Vector Algebra Using î and ĵ.

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Presentation transcript:

Vector Algebra Using î and ĵ

Before now, we described the x and y components of a vector simply as x = ? and y =?. We will start using “unit vectors” with special symbols ĵ and î pronounced “j hat” and “i hat”

Î “ i hat” represents the x- direction Ĵ “j hat” represents the y-direction

Example 1 A rabbit escaping from a fox runs 10.0 m/s at 40.0°north of west. Calculate the rabbits velocity in terms of components and unit vectors. **Setting up a right triangle we should be able to use trig to calculate the vx and Vy

Example 1 cont. Vx is -7.66m/s Vy is 6.43 m/s Therefore the velocity unit vectors can be noted as (-7.66î + 6.43ĵ) m/s

Example 2 A bird flew 100m west then turned 45° north of west and flew 200m. Use algebraic addition of vectors to find the birds net displacement. Vector A is -100îm Vector B is Bx + By = (-141î + 141ĵ)m

Example 2 cont. Vector C (resultant) can be found by adding î and ĵ parts of vectors A and B Vector C is -100îm + (-141î + 141ĵ)m Rearrange so…-100î + (-141î) + 141ĵ Therefore ( -241î + 141ĵ)m are the components

Example 2 cont. C2 = Cx2 + Cy2 C = √ (xî2 + yĵ2) C= √ (-2412 + 1412) C= 279m

Example 2 cont. The angle of displacement is then calculated using: Θ = tan-1 ( Cy / Cx) Θ = tan-1 (yĵ /xî) Θ = tan-1(141/241) = 74° Finally, Vector C is 280m, 30.36° north of west