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6.1 – Vectors in the Plane

6.1 – Vectors in the Plane. What are Vectors?. Vectors are a quantity that have both magnitude (length) and direction , usually represented with an arrow:. This includes force, velocity, and acceleration Component Form : v = <-2,3>. Naming Vectors.

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6.1 – Vectors in the Plane

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  1. 6.1 – Vectors in the Plane

  2. What are Vectors? • Vectors are a quantity that have both magnitude (length) and direction, usually represented with an arrow: • This includes force, velocity, and acceleration • Component Form: • v = <-2,3>

  3. Naming Vectors A vector can also be written as the letters of its head and tail with an arrow above: A – initial point B – terminal point 

  4. Scalars • A quantity with magnitude alone, but no directions, is not a vector, it’s called a scalar • For example, the quantity “60 miles per hours” is a regular number, or scalar. The quantity “60 miles per hour to the northwest” is a vector, because it has both size and direction

  5. Components To do computations with vectors, we place them in the plane and find their components. (5,6) v (2,2)

  6. Components The initial point is the tail, the head is the terminal point. The components are obtained by subtracting coordinates of the initial point from those of the terminal point. (5,6) v (2,2)

  7. Components The first component of v is 5 -2 = 3. The second is 6 -2 = 4. We write v = <3,4> (5,6) v (2,2)

  8. Magnitude of a Vector • The magnitude (or length) of a vector is shown by two vertical bars on either side of the vector: |a| • OR it can be written with double vertical bars: ||a||

  9. Magnitude of a Vector Find the magnitude of the vector: V = <-2,3>

  10. Finding Magnitude of a Vector

  11. Showing Vectors are Equal Let u be the vector represented by the directed line segment from R to S, and v the vector represented by the directed line segment from O to P. Prove that u =v.

  12. Addition • To add vectors, simply add their components. • For example, if v = <3,4> and w = <-2,5>, then v + w = <1,9>.

  13. Multiples of Vectors Given a real number c, we can multiply a vector by c by multiplying its magnitude by c: 2v -2v v Notice that multiplying a vector by a negative real number reverses the direction.

  14. Scalar Multiplication • To multiply a vector by a real number, simply multiply each component by that number. If v = <3,4> and w = <-2,5>, then: • -2v = • 4v – 2w =

  15. Vector Operations Example

  16. Vector Operations Example

  17. Unit Vectors • A unit vector is a vector with magnitude (length) of 1. • Given a vector v, we can form a unit vector by multiplying the vector by 1/||v||. • Or you can think of this as v/||v|| • (The vector divided by its magnitude)

  18. Finding a Unit Vector

  19. Finding a Unit Vector

  20. Standard Unit Vectors • A vector such as <3,4> can be written as 3<1,0> + 4<0,1>. • For this reason, these vectors are given special names: i = <1,0> and j = <0,1>. • A vector in component form v = <a,b> can be written ai + bj. • For example, rewrite the vector <-3, 2>

  21. Direction Angles • The precise way to specify the direction of a vector is to state its direction angle (not its slope). v

  22. Direction Angles

  23. Finding the components of a Vector

  24. Finding the components of a Vector

  25. Examples • Find the component form of v, with magnitude 15 and a direction angle of 40 degrees. • Find the component form of vector v with magnitude 6 and direction angle of 115 degrees.

  26. Examples • Find the component form of v, with magnitude 15 and a direction angle of 40 degrees. <15 cos 40, 15 sin 40> = <11.491, 9.642> • Find the component form of vector v with magnitude 6 and direction angle of 115 degrees. <6 cos 115, 6 sin 115> = <-2.536, 5.438>

  27. Finding the Direction Angle of a Vector

  28. Finding the Direction Angle of a Vector

  29. Finding the direction angle Find the direction angle for the vector <8, -4> v

  30. Velocity and Speed • The velocity of a moving object is a vector because velocity has both magnitude and direction. • The magnitude of velocity is speed.

  31. Word Problem An airplane is flying on a compass heading (bearing) of 170 degrees at 460 mph. A wind is blowing with a bearing of 200 degrees at 80 mph. • Find the component form of the velocity of the airplane • Find the actual ground speed and direction of the plane.

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