- #1
ian_dsouza
- 48
- 3
I am revising vectors at the moment. So, this is not a homework question.
In 2-D cartesian space, I have two vectors, A = <a,2> and B = <1,3>. I want to find the limit of the angle between the vectors as a -> -∞.
Geometrically, I know that direction of vector B approaches 180° as a→-∞. Thus, the angle between the vectors apporaches 180° - atan(3) = 108.4°.
I now want to solve this using more algebraic means. The cosine of the angle θ, between the two vecors is given by,
cosθ = [itex]\frac{\textbf{A . B}}{|\textbf{A}| |\textbf{B}|}[/itex] .
This gives,
cosθ = [itex]\frac{a + 6}{\sqrt{a^{2}+4} \sqrt{10}}[/itex]
In this form, as a→-∞, cosθ should be negative, which makes sense.
To algebraically get the limit of cosθ, I divide numerator and denominator by a, to get,
cosθ = [itex]\frac{1+\frac{6}{a}}{\sqrt{1+\frac{4}{a^{2}}} \sqrt{10}}[/itex]
When you take limits as a→-∞, you get,
[itex]\frac{1+0^{-}}{\sqrt{1+0^{+}} \sqrt{10}}[/itex]
Since the denominator terms were magnitudes of vectors, they have to be positive. This gives cosθ as positive. But this isn't true. In fact it is the negative of that limit.
Can someone help me out with this. I have a feeling that I'm missing something blatantly obvious and would appreciate it you could point it out, if that's the case!
In 2-D cartesian space, I have two vectors, A = <a,2> and B = <1,3>. I want to find the limit of the angle between the vectors as a -> -∞.
Geometrically, I know that direction of vector B approaches 180° as a→-∞. Thus, the angle between the vectors apporaches 180° - atan(3) = 108.4°.
I now want to solve this using more algebraic means. The cosine of the angle θ, between the two vecors is given by,
cosθ = [itex]\frac{\textbf{A . B}}{|\textbf{A}| |\textbf{B}|}[/itex] .
This gives,
cosθ = [itex]\frac{a + 6}{\sqrt{a^{2}+4} \sqrt{10}}[/itex]
In this form, as a→-∞, cosθ should be negative, which makes sense.
To algebraically get the limit of cosθ, I divide numerator and denominator by a, to get,
cosθ = [itex]\frac{1+\frac{6}{a}}{\sqrt{1+\frac{4}{a^{2}}} \sqrt{10}}[/itex]
When you take limits as a→-∞, you get,
[itex]\frac{1+0^{-}}{\sqrt{1+0^{+}} \sqrt{10}}[/itex]
Since the denominator terms were magnitudes of vectors, they have to be positive. This gives cosθ as positive. But this isn't true. In fact it is the negative of that limit.
Can someone help me out with this. I have a feeling that I'm missing something blatantly obvious and would appreciate it you could point it out, if that's the case!