Solving a linear equation with a cross product

In summary, the problem is to solve for vector v in the equation \alpha v + ( a \times v ) = b, where \alpha is a scalar and a and b are fixed vectors. Using dot and cross product operations, the unique solution for v is v=\frac{\alpha^{2}b- \alpha (b \times a) + (b \cdot a)a}{\alpha(\alpha^{2}+|a|^{2}}). To solve this, the first step is to find the component of v along axb and the component of v perpendicular to axb.
  • #1
SP90
23
0

Homework Statement



Suppose v is a vector satisfying:

[itex]\alpha v + ( a \times v ) = b[/itex]

For [itex]\alpha[/itex] a scalar and [itex]a, b[/itex] fixed vectors. Use dot and cross product operations to solve the above for v.

Homework Equations



The unique solutions should be:

[itex]v=\frac{\alpha^{2}b- \alpha (b \times a) + (b \cdot a)a}{\alpha(\alpha^{2}+|a|^{2}})[/itex]

I'm having trouble getting there.

The Attempt at a Solution



I get that [itex]( a \times v ) = b - \alpha v[/itex] and then dotting both sides with a and v gives two identities:

[itex](b - \alpha v) \cdot a = 0[/itex]
[itex](b - \alpha v) \cdot v = 0[/itex]

Which rearrange to give

[itex](b \cdot a) = \alpha (v \cdot a)[/itex]
[itex](b \cdot a) = \alpha (v \cdot v) = \alpha |v|^{2}[/itex]

I don't know where to go from here. There's no inverse cross product and I've tried several different combinations of cross and dot products which lead to dead ends, or are just the same equations as those two I've just derived.

Any help would be much appreciated.
 
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  • #2
Hi SP90! :smile:

You have three vectors, a b and axb, with the third perpendicular to the other two.

So start by finding the component of v along axb, and the component of v perpendicular to axb. :wink:
 

1. How do you solve a linear equation with a cross product?

To solve a linear equation with a cross product, you need to first rewrite the equation in the form of ax + by = c. Then, you can use the cross product property of equality to eliminate the variables and solve for the remaining variable. Finally, plug in the solution to find the value of the other variable.

2. What is the cross product property of equality?

The cross product property of equality states that if two ratios are equal, then their cross products are also equal. In other words, if a/b = c/d, then ad = bc. This property is useful in solving linear equations with fractions or variables on both sides.

3. Can you solve a linear equation with a cross product if it has fractions?

Yes, you can solve a linear equation with a cross product even if it has fractions. The key is to first eliminate the fractions by multiplying both sides of the equation by the lowest common denominator (LCD). This will result in an equation without fractions, which can then be solved using the cross product property of equality.

4. Are there any special cases when solving a linear equation with a cross product?

Yes, there are two special cases to consider when solving a linear equation with a cross product. The first is when the cross product results in a zero on one side of the equation, indicating that there is no solution. The second is when the cross product results in a zero on both sides of the equation, indicating that there are infinitely many solutions.

5. Is there a specific order to follow when solving a linear equation with a cross product?

Yes, there is a specific order to follow when solving a linear equation with a cross product. First, rewrite the equation in the form of ax + by = c. Then, use the cross product property of equality to eliminate the variables and solve for the remaining variable. Finally, plug in the solution to find the value of the other variable. It is important to follow this order to ensure an accurate solution.

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