Dot product / vector magnitudes

I guess it's time to change my signature.In summary, the task at hand is to find ||2V+W||. Using the given values of ||V|| = 2, ||W|| = 3, and the angle between vectors = 120 degrees, we can use the equations Cos120 = V(dot)W / ||V|| ||W||, V(dot)V = ||V|| ^2, and W(dot)W = ||W|| ^2 to solve for ||2V+W||. By squaring ||2V+W|| and using the values of V(dot)V, V(dot)W, and W(dot)W, we arrive at an answer of 13^(1/2).
  • #1
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Homework Statement


||V|| = 2
||W|| = 3
angle between vectors = 120 degrees
V(dot)W = -3

Find ||2V+W||


Homework Equations


Cos120 = V(dot)W / ||V|| ||W||

V(dot)V = ||V|| ^2
W(dot)W = ||W|| ^2



The Attempt at a Solution




instead of solving for ||2V+W||

i instead try to solve for ||2V+W|| ^2

||(2V+W)(2V+W)||

= 4V(dot)V + 4V(dot)W + W(dot)W

=|| 4(2) + 4(-3) + 3||

i add these up and get -1

the book has an answer of 13^(1/2)
 
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  • #2
sorry, reviewing my post i found my mistake.
 
  • #3
If [tex]|\mathbf{V}|=2[/tex], then [tex]\mathbf{V}\cdot \mathbf{V}=4[/tex], with a similar correction to [tex]\mathbf{W}\cdot \mathbf{W}=4[/tex]. Also, [tex]|2\mathbf{V}+\mathbf{W}|[/tex] will always be a positive number.
 
  • #4
pearss said:
...

The Attempt at a Solution

instead of solving for ||2V+W||

i instead try to solve for ||2V+W|| ^2

So far, so good. :)

||(2V+W)(2V+W)||

This line is incorrect! There should be no norm sign there. You should note that:
[tex]\| \mathbf{v} \| ^ 2 = \mathbf{v} \cdot \mathbf{v}[/tex]

= 4V(dot)V + 4V(dot)W + W(dot)W

=|| 4(2) + 4(-3) + 3||

You seem to have forgotten to square it.

-------------------

Whoops... it seems that fzero beats me by 3 minutes. =.="
 

1. What is a dot product?

A dot product, also known as a scalar product, is a mathematical operation performed on two vectors to yield a scalar quantity. It is calculated by multiplying the corresponding components of the two vectors and then summing the results.

2. How is a dot product calculated?

The dot product of two vectors, a and b, is calculated using the formula a · b = |a||b| cos θ, where |a| and |b| represent the magnitudes of the two vectors and θ is the angle between them.

3. What is the significance of the dot product?

The dot product has a number of applications in mathematics and physics. It can be used to calculate the angle between two vectors, determine the projection of one vector onto another, and solve systems of linear equations. It also has geometric interpretations, such as determining if two vectors are perpendicular or parallel.

4. How does the dot product relate to vector magnitudes?

The dot product is directly related to the magnitudes of two vectors. The dot product of two vectors is equal to the product of their magnitudes multiplied by the cosine of the angle between them. This means that the dot product is larger when the vectors are more parallel and smaller when they are more perpendicular.

5. Can the dot product be negative?

Yes, the dot product can be negative. This occurs when the angle between two vectors is greater than 90 degrees, resulting in a negative value for cos θ. In this case, the dot product represents the difference between the two vectors, rather than their similarity.

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