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Vector Subtraction

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You and a friend are trying to position a heavy sculpture out in front of your school. Fortunately, the sculpture is on rollers, so you can move it around easily and slide it into place. While you are applying force to the sculpture, it starts to move. The vectors you and your friend are applying look like this:

However, the sculpture starts to move too far and overshoots where it is supposed to be. You quickly tell your friend to pull instead of push, in effect subtracting her force vector, where before it was being added. Can you represent this graphically?

By the end of this Concept, you'll know how to represent the subtraction of vectors and answer this question.

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Vector Subtraction

Guidance

As you know from Algebra, A - B = A + (-B) . When we think of vector subtraction, we must think about it in terms of adding a negative vector. A negative vector is the same magnitude of the original vector, but its direction is opposite.

In order to subtract two vectors, we can use either the triangle method or the parallelogram method from above. The only difference is that instead of adding vectors A and B , we will be adding A and -B .

Example A

Using the triangle method for subtraction.

Example B

Example C

Vocabulary

Negative Vector: A negative vector is a vector that is the same in magnitude as the original vector, but opposite in direction.

Triangle Method: The triangle method is a method of adding vectors by connecting the tail of one vector to the head of another vector.

Guided Practice

1.For the vector subtraction below, make a diagram of the subtraction. \vec{a} - \vec{d}

2.For the vector subtraction below, make a diagram of the subtraction. \vec{b} -  \vec{a}

3. For the vector subtraction below, make a diagram of the subtraction. \vec{d} - \vec{c}

Solutions:

1.

2.

3.

Concept Problem Solution

As you've seen in this Concept, subtracting a vector is the same as adding the negative of the original vector. This is exactly like the rule for adding a negative number to a positive number. Therefore, to change your friend's force vector to a subtraction instead of an addition, you need to change the direction by 180^\circ while keeping the magnitude the same. The graph looks like this:

Practice

\vec{a} is in standard position with terminal point (1, 5) and \vec{b} is in standard position with terminal point (4, 2).

  1. Find the coordinates of the terminal point of \vec{a} - \vec{b} .
  2. What is the magnitude of \vec{a} - \vec{b} ?
  3. What is the direction of \vec{a} - \vec{b} ?

\vec{c} is in standard position with terminal point (4, 3) and \vec{d} is in standard position with terminal point (2, 2).

  1. Find the coordinates of the terminal point of \vec{c} - \vec{d} .
  2. What is the magnitude of \vec{c} - \vec{d} ?
  3. What is the direction of \vec{c} - \vec{d} ?

\vec{e} is in standard position with terminal point (3, 2) and \vec{f} is in standard position with terminal point (-1, 2).

  1. Find the coordinates of the terminal point of \vec{e} - \vec{f} .
  2. What is the magnitude of \vec{e} - \vec{f} ?
  3. What is the direction of \vec{e} - \vec{f} ?

\vec{g} is in standard position with terminal point (5, 5) and \vec{h} is in standard position with terminal point (4, 2).

  1. Find the coordinates of the terminal point of \vec{g} - \vec{h} .
  2. What is the magnitude of \vec{g} - \vec{h} ?
  3. What is the direction of \vec{g} - \vec{h} ?

\vec{i} is in standard position with terminal point (1, 5) and \vec{j} is in standard position with terminal point (-3, 1).

  1. Find the coordinates of the terminal point of \vec{i} - \vec{j} .
  2. What is the magnitude of \vec{i} - \vec{j} ?
  3. What is the direction of \vec{i} - \vec{j} ?

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