Dot Product Question: How to Solve (2a-5b)dot(b+3a) with Unit Vectors?

In summary, the conversation discusses using given unit vectors a and b to solve for a dot product and evaluate an expression. The dot product is associative, distributive, and commutative, and the equations |a + b| = sqrt(3) and (a + b) \cdot (a + b) = 3 are used to find the final answer of -11/2. The unit vectors a and b have a dot product of 1, and this is used in the calculation.
  • #1
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Homework Statement



I'm really at a loss here, if anyone could help me out I'd really appreciate it.


Given 'a' and 'b' unit vectors,

if |a+b| = root3, determine (2a-5b)dot(b+3a)
 
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  • #2
|a + b|^2 = (a + b)[tex]\bullet[/tex](a + b)
and |a + b| = sqrt(3) ==> |a + b|^2 = 3

Now, use the fact that the dot product is associative, distributive, and commutative and the two equations above to see if you can evaluate (2a - 5b) )[tex]\bullet[/tex] (b + 3a).
 
  • #3
well I end up getting 13ab + 6a^2 - 5b^2

The answer is -11/2

I just can't seem to figure out how to get there :s
 
  • #4
Work with (a + b) [itex]\cdot [/itex] (a + b) = 3. You also know that a and b are unit vectors, which means that a [itex]\cdot [/itex] a = 1 and b [itex]\cdot [/itex] b = 1.
 
  • #5
Would I do like

(a+b)dot(a+b)=3

1 + 2ab + 1 = 3

ab = 1/2

then sub 1/2 into the ab and then get 6.5 + 6a^2 - 5b^2 and solve from there?
 
  • #6
Sort of, except that what you show as 6a^2 and -5b^2 is really 6a[itex] \cdot [/itex] a and -5b[itex] \cdot [/itex] b.
 

FAQ: Dot Product Question: How to Solve (2a-5b)dot(b+3a) with Unit Vectors?

1. What is a dot product and how is it calculated?

A dot product is a mathematical operation that takes two vectors and returns a scalar value. It is calculated by multiplying the corresponding components of the two vectors and then adding the results together.

2. Why are unit vectors used in dot product calculations?

Unit vectors are used in dot product calculations because they have a magnitude of 1 and are parallel to the original vector. This makes it easier to calculate the dot product and interpret its meaning.

3. How do you solve a dot product with unit vectors?

To solve a dot product with unit vectors, you first need to determine the components of the two vectors being multiplied. Then, multiply the corresponding components and add the results together. The final answer will be a scalar value.

4. What is the significance of the dot product in vector operations?

The dot product has several important applications in vector operations. It can be used to determine the angle between two vectors, find the projection of one vector onto another, and calculate the work done by a force.

5. Can the dot product of two vectors be negative?

Yes, the dot product of two vectors can be negative. This means that the two vectors are pointing in opposite directions. If the dot product is positive, the vectors are pointing in the same direction, and if it is zero, the vectors are perpendicular to each other.

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