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2.1 Vectors in the plane  (Page 3/29)

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Definition

The sum of two vectors v and w can be constructed graphically by placing the initial point of w at the terminal point of v . Then, the vector sum, v + w , is the vector with an initial point that coincides with the initial point of v and has a terminal point that coincides with the terminal point of w . This operation is known as vector addition    .

This image has two figures. The first has two vectors, v and w with the same initial point. A parallelogram is formed by sketching broken lines parallel to the two vectors. A diagonal line is drawn from the same initial point to the opposite corner. It is labeled “v + w.” The second has two vectors, v and w. Vector v begins at the terminal point of vector w. A parallelogram is formed by sketching broken lines parallel to the two vectors. A diagonal line is drawn from the same initial point as vector w to the opposite corner. It is labeled “v + w.”
(a) When adding vectors by the triangle method, the initial point of w is the terminal point of v . (b) When adding vectors by the parallelogram method, the vectors v and w have the same initial point.

It is also appropriate here to discuss vector subtraction. We define v − w as v + ( − w ) = v + ( −1 ) w . The vector v − w is called the vector difference    . Graphically, the vector v − w is depicted by drawing a vector from the terminal point of w to the terminal point of v ( [link] ).

This image has two figures. The first figure has two vectors, one labeled “v” and the other labeled “w.” Both vectors have the same initial point. A third vector is drawn between the terminal points of v and w. It is labeled “v – w.” The second figure has two vectors, one labeled “v” and the other labeled “-w.” The vector “-w” has its initial point at the terminal point of “v.” A parallelogram is created with broken lines where “v” is the diagonal and “w” is the top side.
(a) The vector difference v − w is depicted by drawing a vector from the terminal point of w to the terminal point of v . (b) The vector v − w is equivalent to the vector v + ( − w ) .

In [link] (a), the initial point of v + w is the initial point of v . The terminal point of v + w is the terminal point of w . These three vectors form the sides of a triangle. It follows that the length of any one side is less than the sum of the lengths of the remaining sides. So we have

‖ v + w ‖ ≤ ‖ v ‖ + ‖ w ‖ .

This is known more generally as the triangle inequality    . There is one case, however, when the resultant vector u + v has the same magnitude as the sum of the magnitudes of u and v . This happens only when u and v have the same direction.

Combining vectors

Given the vectors v and w shown in [link] , sketch the vectors

  1. 3 w
  2. v + w
  3. 2 v − w
    This figure has two vectors. They are vector v and vector w. They are not connected.
    Vectors v and w lie in the same plane.
  1. The vector 3 w has the same direction as w ; it is three times as long as w .
    This figure has two vectors. The first is labeled “w.” The second one is parallel to “w” and is labeled “3w.” It is three times as long as w in the same direction.
    Vector 3 w has the same direction as w and is three times as long.
  2. Use either addition method to find v + w .
    This image has two figures. The first has two vectors, labeled “v” and “w.” They both have the same initial point. A third vector is drawn, labeled “v + w.” It is the diagonal of the parallelogram formed by having sides parallel to vectors v and w. The second figure is a triangle formed by having vector v on one side and vector w adjacent to v. The terminal point of v is the initial point of w. The third side is labeled “v + w.”
    To find v + w , align the vectors at their initial points or place the initial point of one vector at the terminal point of the other. (a) The vector v + w is the diagonal of the parallelogram with sides v and w (b) The vector v + w is the third side of a triangle formed with w placed at the terminal point of v .
  3. To find 2 v − w , we can first rewrite the expression as 2 v + ( − w ) . Then we can draw the vector − w , then add it to the vector 2 v .
    This figure is a triangle formed by having vector 2v on one side and vector -w adjacent to 2v. The terminal point of 2v is the initial point of -w. The third side is labeled “2v - w.”
    To find 2 v − w , simply add 2 v + ( − w ) .
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Using vectors v and w from [link] , sketch the vector 2 w − v .


This figure is a triangle formed by having vector 2w on one side and vector -v adjacent to 2w. The terminal point of 2w is the initial point of -v. The third side is labeled “2w – v.”

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Vector components

Working with vectors in a plane is easier when we are working in a coordinate system. When the initial points and terminal points of vectors are given in Cartesian coordinates, computations become straightforward.

Comparing vectors

Are v and w equivalent vectors?

  1. v has initial point ( 3 , 2 ) and terminal point ( 7 , 2 )
    w has initial point ( 1 , −4 ) and terminal point ( 1 , 0 )
  2. v has initial point ( 0 , 0 ) and terminal point ( 1 , 1 )
    w has initial point ( −2 , 2 ) and terminal point ( −1 , 3 )
  1. The vectors are each 4 units long, but they are oriented in different directions. So v and w are not equivalent ( [link] ).
    This figure is a Cartesian coordinate system with two vectors. The first vector labeled “v” has initial point at (3, 2) and terminal point (7, 2). It is parallel to the x-axis. The second vector is labeled “w” and has initial point (1, -4) and terminal point (1, 0). It is parallel to the y-axis.
    These vectors are not equivalent.
  2. Based on [link] , and using a bit of geometry, it is clear these vectors have the same length and the same direction, so v and w are equivalent.
    This figure is a Cartesian coordinate system with two vectors. The first vector labeled “v” has initial point at (0, 0) and terminal point (1, 1). The second vector is labeled “w” and has initial point (-2, 2) and terminal point (-1, 3).
    These vectors are equivalent.
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Source:  OpenStax, Calculus volume 3. OpenStax CNX. Feb 05, 2016 Download for free at http://legacy.cnx.org/content/col11966/1.2
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