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Operations with Vectors

Resultant vectors based on adding or subtracting vectors or multiplying by a constant.

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Operations with Vectors

When two or more forces are acting on the same object, they combine to create a new force.  A bird flying due south at 10 miles an hour in a headwind of 2 miles an hour only makes headway at a rate of 8 miles per hour.  These forces directly oppose each other.  In real life, most forces are not parallel.  What will happen when the headwind has a slight crosswind as well, blowing NE at 2 miles per hour.  How far does the bird get in one hour?

Watch This

http://www.youtube.com/watch?v=EYIxFJXoUvA James Sousa: Vector Operations

Guidance

When adding vectors, place the tail of one vector at the head of the other.  This is called the tail-to-head rule The vector that is formed by joining the tail of the first vector with the head of the second is called the resultant vector .

Vector subtraction reverses the direction of the second vector. \overrightarrow{a}-\overrightarrow{b}=\overrightarrow{a}+(-\overrightarrow{b}) :

Scalar multiplication means to multiply a vector by a number.  This changes the magnitude of the vector, but not its direction.  If \overrightarrow{v}=<3,4 > , then 2 \overrightarrow{v}=< 6,8 > .

Example A

Two vectors \overrightarrow{a}  and \overrightarrow{b} , have magnitudes of 5 and 9 respectively.   The angle between the vectors is 53^\circ .  Find \big |\overrightarrow{a}+\overrightarrow{b} \big | .   

Solution:  Adding vectors can be done in either order (just like with regular numbers).  Subtracting vectors must be done in a specific order or else the vector will be negative (just like with regular numbers).  In either case, use geometric reasoning and the law of cosines with the parallelogram that is formed to find the magnitude of the resultant vector. 

In order to fine the magnitude of the resulting vector (x) , note the triangle on the bottom that has sides 9 and 5 with included angle 127^\circ

x^2 &= 9^2+5^2-2 \cdot 9 \cdot 5 \cdot \cos 127^\circ\\x & \approx 12.66

Example B

Using the picture from Example A, what is the angle that the sum \overrightarrow{a}+\overrightarrow{b}  makes with \overrightarrow{a}

Solution:  Start by drawing a good picture and labeling what you know.  \big |\overrightarrow{a} \big |=5, \big |\overrightarrow{b} \big |=9, \big |\overrightarrow{a}+\overrightarrow{b} \big | \approx 12.66 .  Since you know three sides of the triangle and you need to find one angle, this is the SSS application of the Law of Cosines.

9^2 &= 12.66^2+5^2-2 \cdot 12.66 \cdot 5 \cdot \cos \theta\\\theta &= 34.6^\circ

Example C

Elaine started a dog walking business.  She walks two dogs at a time named Elvis and Ruby.  They each pull her in different directions at a 45^\circ  angle with different forces.  Elvis pulls at a force of  25 \ N and Ruby pulls at a force of 49 \ N .  How hard does Elaine need to pull so that she can stay balanced?  Note:  N stands for Newtons which is the standard unit of force.


Solution:  Even though the two vectors are centered at Elaine, the forces are added which means that you need to use the tail-to-head rule to add the vectors together.  Finding the angle between each component vector requires logical use of supplement angles

x^2 &= 49^2+25^2-2 \cdot 49 \cdot 25 \cdot \cos 135^\circ\\x & \approx 68.98 \ N

In order for Elaine to stay balanced, she will need to counteract this force with an equivalent force of her own in the exact opposite direction. 

Concept Problem Revisited

A bird flying due south at 10 miles an hour with a cross headwind of 2 mph heading NE would have a force diagram that looks like this:

The angle between the bird’s vector and the wind vector is 45^\circ  which means this is a perfect situation for the Law of Cosines.  Let x=  the red vector.

x^2 &= 10^2+2^2-2 \cdot 10 \cdot 2 \cdot \cos 45^\circ\\x &\approx 8.7

The bird is blown slightly off track and travels only about 8.7 miles in one hour. 

Vocabulary

A resultant vector is the vector that is produced when two or more vectors are summed or subtracted.  It is also what is produced when a single vector is scaled by a constant. 

Scaling a vector means that the components are each multiplied by a common scale factor.  For example:  4 \cdot < 3,2 > = < 12, 8 >

Guided Practice

1. Find the magnitude of \big |\vec{a}-\vec{b} \big |  from Example A. 

2. Consider vector \overrightarrow{v}= < 2, 5 >  and vector \overrightarrow{u}= < -1, 9 > .  Determine the component form of the following: 3 \overrightarrow{v}-2 \overrightarrow{u}

3. An airplane is flying at a bearing of 270^\circ  at 400 mph.  A wind is blowing due south at 30 mph.  Does this cross wind affect the plane’s speed? 

Answers:

1. 

The angle between -\overrightarrow{b}  and \overrightarrow{a}  is 53^\circ  because in the diagram \overrightarrow{b}  is parallel to -\overrightarrow{b}  so you can use the fact that alternate interior angles are congruent.  Since the magnitudes of vectors \overrightarrow{a}  and -\overrightarrow{b}  are known to be 5 and 9, this becomes an application of the Law of Cosines.

y^2 &= 9^2+5^2-2 \cdot 9 \cdot 5 \cdot \cos 53^\circ\\y & \approx 7.2\\

2. Do multiplication first for each term, followed by vector subtraction.

3 \cdot \overrightarrow{v}-2 \cdot \overrightarrow{u} &= 3 \cdot < 2, 5 > -2 \cdot < -1, 9 > \\&= < 6, 15 > - < - 2, 18 > \\&= < 8, -3 >

3. Since the cross wind is perpendicular to the plane, it pushes the plane south as the plane tries to go directly east.  As a result the plane still has an airspeed of 400 mph but the groundspeed (true speed) needs to be calculated.

400^2 + 30^2 &= x^2\\x & \approx 401

Practice

Consider vector \overrightarrow{v}=< 1, 3 >  and vector \overrightarrow{u}= < -2, 4 >

  1. Determine the component form of 5 \overrightarrow{v}-2 \overrightarrow{u}
  2. Determine the component form of -2 \overrightarrow{v}+4 \overrightarrow{u}
  3. Determine the component form of 6 \overrightarrow{v}+\overrightarrow{u}
  4. Determine the component form of 3 \overrightarrow{v}-6 \overrightarrow{u}
  5. Find the magnitude of the resultant vector from #1.
  6. Find the magnitude of the resultant vector from #2.
  7. Find the magnitude of the resultant vector from #3.
  8. Find the magnitude of the resultant vector from #4.
  9. The vector <3, 4 >  starts at the origin.  What is the direction of the vector?
  10. The vector <-1, 2 >  starts at the origin.  What is the direction of the vector?
  11. The vector < 3, -4 >  starts at the origin.  What is the direction of the vector?
  12. A bird flies due south at 8 miles an hour with a cross headwind blowing due east at 15 miles per hour.  How far does the bird get in one hour? 
  13. What direction is the bird in the previous problem actually moving?
  14. A football is thrown at 50 miles per hour due north.  There is a wind blowing due east at 8 miles per hour.  What is the actual speed of the football?
  15. What direction is the football in the previous problem actually moving?

Vocabulary

component form

component form

Component form is used to describe vectors algebraically in terms of their x and y components.
law of cosines

law of cosines

The law of cosines is a rule relating the sides of a triangle to the cosine of one of its angles. The law of cosines states that c^2=a^2+b^2-2ab\cos C, where C is the angle across from side c.
Magnitude

Magnitude

The magnitude of a line segment or vector is the length of the line segment or vector.
Resultant

Resultant

A resultant is a vector representing the sum of two or more vectors.
scalar multiplication

scalar multiplication

Scalar multiplication means to multiply a vector by a number.
scaling a vector

scaling a vector

Scaling a vector means that the components of the vector are each multiplied by a common scale factor.
tail-to-head rule

tail-to-head rule

The tail-to-head rule tells you to add two vectors by placing the tail of one vector at the head of the other vector.
Vector

Vector

A vector is a mathematical quantity that has both a magnitude and a direction.

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