Thread: Two vector problems I need some help with

1. Two vector problems I need some help with

1. Determine whether the vectors [2,-1] and [-2,-5] are orthogonal.

I remember learning this stuff last year and I think it has something to do with the dot product of vectors maybe? Help would be greatly appreciated.

2. Find the angle between vector u = [-2, 5] and vector v= [-1, 3].

Once again, I completely forget how to do this. Cross product?

2. Originally Posted by Jph93
1. Determine whether the vectors [2,-1] and [-2,-5] are orthogonal.
2. Find the angle between vector u = [-2, 5] and vector v= [-1, 3].
If the dot product is zero, $[2,-1]\cdot [-2,-5]=?$ they are orthogonal.

The angle between $u~\&~v$ is $\arccos \left( {\frac{{u \cdot v}}{{\left\| u \right\|\left\| v \right\|}}} \right)$

3. So -4+5= 1
not orthogonal

and

17/(5.39)(3.16)
= 17/17
=1
arccos1=0

so 0 degrees?

4. Originally Posted by Jph93
17/(5.39)(3.16)
= 17/17
=1
arccos1=0
You have missed the lengths.
$\left\| u \right\| = \sqrt {29} \;\& \,\left\| v \right\| = \sqrt {10}$

5. They equal 17 when multiplied right?

6. Originally Posted by Jph93
They equal 17 when multiplied right?
Oh NO!
$\sqrt {10} \cdot \;\sqrt {29} \ne 17\;,\,17^2 = 289 \ne 290$

7. Originally Posted by Plato
If the dot product is zero, $[2,-1]\cdot [-2,-5]=?$ they are orthogonal.

The angle between $u~\&~v$ is $\arccos \left( {\frac{{u \cdot v}}{{\left\| u \right\|\left\| v \right\|}}} \right)$
I am still re-learning my Algebra and have not started on pre-calc yet, but from what I remember about vectors, this notation [2,-1] is the distance of the vector from the origin in the x and y directions, correct?

If this is the case, could you not also solve this problem by calculating then comparing the slope of both vectors as lines?

ie, $m = \frac{y_2 - y_1}{x_2 - x_1}, \,$

$m_1 = \frac{-1 - 0}{2 - 0} = \frac{-1}{2}, \,$

$m_2 = \frac{-5 - 0}{-2 - 0} = \frac{5}{2}$

since $m_1 \neq \frac{-1}{m_2}$, vectors are not orthogonal.

Is this approach correct?

8. Well, that is correct as far as it goes.
But notice that the OP is about vectors, whereas you have used traditional analytical geometry.

9. Originally Posted by Plato
If the dot product is zero, $[2,-1]\cdot [-2,-5]=?$ they are orthogonal.

The angle between $u~\&~v$ is $\arccos \left( {\frac{{u \cdot v}}{{\left\| u \right\|\left\| v \right\|}}} \right)$
First one not orthogonal because the dot product is not 0?

So Plato, is the answer for the angle one $0.05875$ radians or $3.37$ degrees?

Ohh I know this ain't my problem but I stumbled across it and found it good practise on recapping my knowledge on these things...

10. Originally Posted by dwat
, is the answer for the angle one $0.05875$ radians or $3.37$ degrees?
That is approx correct.
If the angle between two vectors is $0\text{ or }\pi$ then the vectors must be multiples of one another.
That is clearly not the case here.