Vectors in the Plane

1
Vectors in the Plane

• 8.4

2
Scalar

• A single real number that indicates magnitude or
  size of a given quantity. Temperature, distance
  height, area, volume can be represented by
  scalars.

3
Directed Line Segments

• Used to represent a magnitude and a direction or
  quantities such as force, velocity, and
  acceleration.

4
Directed Line Segments

• Named just like a ray in geometry. The initial
  point (P) is listed first, then the terminal
  point (Q). PQ

5
Magnitude or Length of Directed Line Segments

• PQ is the symbol for the magnitude. The
  distance formula will be used to find this value.
• D v(x2-x1)2 (y2-y1)2

6
Equivalent Directed Line Segments

• Two directed line segments with the same length
  and direction.

7
Vectors

• The set of all directed line segments equivalent
  to the directed line segment PQ is the vector v
  PQ.

8
Equal Vectors

• Two vectors are equal if their corresponding
  directed line segments are equivalent.

9
Example

• Showing that vectors are equal
• Show that the magnitude of each is equal.
• Show that the slopes of the vectors are equal.
• Note the vectors are not in the same spot, yet
  they will be equivalent.

10
Proving Vectors Equal

• R (7,-3), S (4,-5), O (0,0), and P
  (-3,-2). Prove that u v where u is the vector
  represented by the directed line segment RS and v
  is the vector represented by the directed line
  segment OP.

11
Component Form of a Vector

• If v is a vector in the plane equal to the vector
  with initial point (x1,y1) and terminal point
  (x2,y2), then the component form of v is
• vlt(x2x1), (y2y1)gtltv1,v2gt

12
Vector Operations

• Vector Addition
• uvltu1,u2gtltv1,v2gt
• ltu1v1,u2v2gt
• Scalar Multiplication
• ku kltu1,u2gt ltku1,ku2gt

13
Parallelogram Law for Vector Addition

• u v can be obtained geometrically by joining
  the terminal point of the first vector to the
  initial point of the second vector. The directed
  line segment that joins the initial point of the
  first and the terminal point of the second is
  called the sum.

14
Parallelogram Law for Vector Addition
v
u
u v
15
Magnitude or Length of a Vector

• Let vlt(x2x1), (y2y1)gtltv1,v2gt The distance
  formula will be used to find this value if given
  P and Q where vPQ.
• v v(x2-x1)2 (y2-y1)2
• v(v1)2 (v2)2

16
Unit Vector

• A vector with length one that is in the direction
  of the original vector. To find the unit vector,
  take the vector and divide each of its components
  by its magnitude.
• u v/v

17
Write the Linear Combination of the standard unit
vectors i and j

• Given a vector v
• V (v cos ?)i (vsin ? )j
• The unit vector in the direction of v is
• U v/v (cos ?)i (sin ? )j

18
Direction Angles

• The angle used to specify the direction of the
  vector.
• The angle is the one that the vector makes with
  the positive x-axis.
• Using sohcahtoa, we can determine that
• vcos ? gives the horizontal component and
  v sin ? gives the vertical component.

19
Applications

• Velocity is a vector because it has a magnitude
  and direction. The magnitude is called the
  speed.
• BearingAn angle measured from straight north in
  the clockwise direction.
• The new velocity or force vector will be the sum
  of the two given.

20
Applications

• The actual direction of the force or the velocity
  can be found by taking the arctan(y/x). You may
  have to add 180 to this result depending on the
  quadrant that your resultant vector is in.