How to solve quadratic vector equation?

[Jeno]

Homework Statement

(a)² + a · b - c = 0

a and b are vectors which (in general) with different directions while c is a scalar. How to solve for a?​

The Attempt at a Solution

(a)² + a · b - c = 0
(a + (1/2)b)² - (1/4)b² - c = 0
(a + (1/2)b)² = (1/4)b² + c​

In this case, if i carry the square from left hand side to right hand side, it appears that the left hand side turns into a vector while the right hand side remains as a scalar. I wonder why this happens and what would be the proper way to solve this equation.​

 

[LCKurtz]

What does a² mean if a is a vector?

 

[Jeno]

What does a² mean if a is a vector?

Hi, I think a² should be a scalar with the magnitude of the square of the length of the vector a. Is there any problem with it? Do you know how to solve this equation?

Thank you.

 

[Mark44]

Hi, I think a² should be a scalar with the magnitude of the square of the length of the vector a. Is there any problem with it? Do you know how to solve this equation?

Thank you.

I'm pretty certain that what you have as a² should be written as a · a. Assuming your equation is a · a + a · b - c = 0, you can use one of the definitions of the dot product to rewrite this equation as
|a|² + |a| |b|cos(theta) - c = 0, where theta is the angle between a and b. This equation is quadratic in |a|.

 

[LCKurtz]

Given that a² means a$\cdot$a the equation doesn't have a unique solution and may have none. If you let:

$$\vec a = \langle x,y,z\rangle,\, \vec b = \langle u,v,w\rangle$$

the equation becomes:

$$x^2+y^2+z^2+xu+yv+zw-c = 0$$

which, depending on the various constants, might be a sphere or nothing.

 

[Jeno]

Ya, i think a² = a · a . Is there any other definition for a²?

So, if i don't know the direction of a, then i can't solve for a is it?

I think, in my situation, i can guess the direction for a. Let's say if i know that the direction of a is r (a vector with magnitude of r, such that r/r is a unit vector), so how can i solve for the equation without expressing it in the angle between them (theta), and just the vectors and scalar of b, c and r?

a=-(r/2r²)( +- (4r²c + (b · r)² )^1/2 + r · u )

This is the answer given in the book. It seems like after applying the formula to find the root of quadratic equation, then he multiply it with r²/r². But it didn't sound quite correct to me if you consider the vector BAC CAB rule. The cross product term didn't appear in this equation, and in my situation, b can be in any direction, which doesn't restricted to be parallel to a.

which, depending on the various constants, might be a sphere or nothing.

Sorry, what do you mean by "nothing" here?

Thank you.

 

[LCKurtz]

Sorry, what do you mean by "nothing" here?

Thank you.

I mean there is no solution. For example, with the right choice for u,v,w, and c, the equation might end up something like

$$(x-1)^2 + (y+2)^2+(z-3)^2 = -10$$

having no real solutions.

 

[Jeno]

So is it similar to the normal quadratic equation where b² - 4ac < 0 gives no real solution? ok, then i understand.

But i still don't understand how the book get the the expression above. I think the expression above should be valid, since it is written on the book, but i just don't know how to get that through a proper way.

Can anybody help? Thank you.

 

[HallsofIvy]

What "expression above"? You showed no expression that you said came from the book.

 

[Jeno]

What "expression above"? You showed no expression that you said came from the book.

a=-(r/2r2)( +- (4r2c + (b · r)2 )1/2 + r · u )

This is the expression i meant. Thank you.

 

[Mark44]

a=-(r/2r2)( +- (4r2c + (b · r)2 )1/2 + r · u )

This is the expression i meant. Thank you.

How is this equation related to the equation in post #1? The original equation involved vectors a and b, and a constant c. The equation you just showed has r and u, which aren't in the original equation.

Also, your last equation is very difficult to read, as it has what appear to be exponents that aren't shown as such. For example what you have written as 4r2c could be (4r)(2c) or might be 4r²c. And I'm guessing that (b · r)2 )1/2 is (b · r)2 )^1/2.

It's pretty easy to write exponents using the advanced menu. Click the Go advanced button, and use the X² button.

 

[Jeno]

How is this equation related to the equation in post #1? The original equation involved vectors a and b, and a constant c. The equation you just showed has r and u, which aren't in the original equation.

Also, your last equation is very difficult to read, as it has what appear to be exponents that aren't shown as such. For example what you have written as 4r2c could be (4r)(2c) or might be 4r²c. And I'm guessing that (b · r)2 )1/2 is (b · r)2 )^1/2.

It's pretty easy to write exponents using the advanced menu. Click the Go advanced button, and use the X² button.

Sorry, i have typed it wrong. The expression should be as follow:

a=-(r/2r²)( +- (4r²c + (b · r)² )^1/2 + r · b )

where r is a vector with direction of a and length of r.

Thank you.