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Operations with Vectors
Introduction: In this lesson, vectors will be added and subtracted. We will also describe scalar multiplication of vectors.

The Lesson:

Vectors can be added or subtracted as well as multiplied by a scalar (number). Addition and subtraction result in another vector for which magnitude and direction are both changed. Scalar multiplication does not affect direction, but does change the magnitude unless the scalar is 1 because and v remain unchanged under scalar multiplication by 1.
Scalar multiplication:
Suppose v is the vector (-3, 2). How are its magnitude and direction affected when it is multiplied by the scalar 5?
  1. The direction of v, which in standard position is in quadrant 2, is found from .
    Since v is in quadrant 2, its direction is 180º - 33.69º = 146.31º.
    Or alternatively, N by 90º - 33.69º = 56.31º W
  2. Multiplication of v by 5, which is written 5v, gives us a new vector:
  3. The direction is unaffected since this is still a quadrant 2 vector in standard position. The direction is given by , the same angle we calculated for the original vector v.


  4. The magnitude of 5v is .
    Notice that this is where .


Generalizing, when we multiply a given vector w by a scalar (number) a,
  1. The magnitude is also multiplied by a factor of a: .
  2. The direction does not change.

Addition and Subtraction of Vectors:
To add or subtract two vectors whose components are known, we simply add or subtract the components. Therefore, if v = (3, -4) and w = (5, 9) we have:
  • v - w = (3 - 5, -4 - 9) = (-2, -13)
  • v + w = (3 + 5, -4 + 9) = (8, 5)
In each of these cases, both the magnitude and direction are changed by adding and subtracting. Notice that


  • The magnitude of v - w = (-2, -13) is
    .


  • The magnitude of v + w = (8, 5) is
    .
Before considering the directions of v - w and v + w we need to first determine the directions of each vector individually.
The direction of v, a 4th quadrant vector, is found from which can be stated as either or .


The direction of w, a quadrant 1 vector, is found from .
  • v - w = (-2, -13) has direction given by . Since this is a quadrant 3 vector, the direction is .


  • v + w = (8, 5) has direction given by . Since this is a quadrant 1 vector, is the direction of v + w.
In the next section, we will diagram both the addition and subtraction of these two vectors.

Notice that v + w is the diagonal of a parallelogram formed by v and w. We translate the vector v so that the tail (initial point) of v is at the head (terminal point) of w. We connect the tail of w to the head of v to diagram v + w and it is the diagonal of a parallelogram formed by v and w. This is called "head-to-tail" addition of vectors.

To subtract w - v we first diagram -v = (-3, 4) and translate -v so that the tail of v connects with the head of w. In other words, we add w + (-v). Drawing from the tail of w to the head of the translated -v we construct w - v = (2, 13). Notice this is in the exact opposite direction of v - w =(-2, -13).

We would calculate the magnitude of w - v as and its direction would be found using .


Generalizing:
  1. When adding two vectors v and w, we add the components and recalculate the magnitude and direction.
  2. In a diagram, we place the tail of v to the head of w. Then we draw v + w by connecting the tail of w to the head of v.
  3. When subtracting v from w, we add w + (-v) as in (i) and (ii) generalizations above.
Let's Practice:
  1. If v = (1, -2) and w = (5, 4) find the components, magnitude, and direction of both v + w and v - w.
Components:
  • v + w = (1 + 5, -2 + 4) = (6, 2)
  • v - w = (1 - 5, -2 - 4) = (-4, -6).
Magnitudes:
Directions:
  • For v + w, we use
  • For v - w, we use . Since v - w = (-4, -6) is a quadrant 3 vector, the direction would be given by .
  1. A plane is traveling 30º E of North at 500 mph when a wind of 37 mph from 30º S of West starts to affect the flight. What will be the magnitude and direction of the plane’s velocity if it does not correct for the wind?
We calculate the components of the velocity vectors of both the plane and the wind:
  • Plane: p =
  • Wind: w =
The velocity vector of the plane affected by the wind is p + w = (283, 451.5).
  • The magnitude of p + w is given by mph.
The direction is found from .
  • This would be 58º N of East or 32º E of North.
A diagram is shown below adding the velocity of the plane and that of the wind.


Examples
Example If v = (1, 2) and w = (-4, 5) find the components, magnitude, and direction of both v + w and v - w.
What is your answer?
 
Example A force of 500 pounds is required to pull a block up a ramp inclined at 15º with the horizontal. Find the weight of the block.
What is your answer?
 
Example Forces with magnitudes of 1000 newtons and 700 newtons act on an object at angles of 15º and -35º respectively with the horizontal. Find the direction and magnitude of the resultant of these forces.
What is your answer?
 



M Ransom

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