c4 vectors angles

alevels2k17

14

in the dot product rule .. when we find the value for the cos(theta) is the angle always acute????????

Original post by pondsteps

in the dot product rule .. when we find the value for the cos(theta) is the angle always acute????????

Not necessarily, you can have cosine equal to a negative number which would give you an obtuse angle.

Reply 2

alevels2k17

OP

14

but isnt the angle between the two lines that have intersected an acute angle?? im confused

Original post by pondsteps

but isnt the angle between the two lines that have intersected an acute angle?? im confused

Two lines which intersect have an acute angle and an obtuse one. When you carry out the working for the angle between two vectors, the angle found is always one such that the vector directions are tail-to-tail or head-to-head.

For example, dotting these two directions will give the acute angle:

Spoiler

But dotting these two directions will give the obtuse angle:

Spoiler

Clearly, the direction of the blue line is opposite from the two pics, meaning a factor of -1 for it, therefore the sign of cosine would differ for the two scenarios, thus you can get an obtuse and acute angles.

(edited 6 years ago)

OP

thanks a bunch!! :h:

Original post by pondsteps

but isnt the angle between the two lines that have intersected an acute angle?? im confused

There will generally be two angles between vectors, an acute angle and an obtuse angle.

\theta

in the dot product formula is the angle between two vectors that are pointing away from each other or towards each other.

So in this diagram the dot product formula will give you the blue obtuse angle. If you want the red acute angle then you could just subtract from 180 or use the formula

\displaystyle \cos \theta = \left|\frac{\mathbf{a}\cdot \mathbf{b}}{|\mathbf{a}||\mathbf{b}|}\right|

This formula always gives you the acute angle between two vectors.

(edited 6 years ago)

Reply 6

alevels2k17

OP

14

Original post by notnek

There will generally be two angles between vectors, an acute angle and an obtuse angle.

\theta

in the dot product formula is the angle between two vectors that point in the same direction.

So in this diagram the dot product formula will give you the blue obtuse angle. If you want the red acute angle then you could just subtract from 180 or use the formula

\displaystyle \cos \theta = \left|\frac{\mathbf{a}\cdot \mathbf{b}}{|\mathbf{a}||\mathbf{b}|}\right|

This formula always gives you the acute angle between two vectors.

doesnt the one in bold contradict what u said before it?? :frown:

the formula always gives the acute angle between two vectors then how did it give us the obtuse angle in the top example?

Original post by pondsteps

doesnt the one in bold contradict what u said before it?? :frown:

the formula always gives the acute angle between two vectors then how did it give us the obtuse angle in the top example?

It does not contradict Evidently, that is not the exact formula you are given in the book (at least I don't think) so the twist here is that you are taking the modulus of whatever the dot product over magnitudes is, so the value of cosine is always positive thus the angle you calculate in the end is acute. Then to get the obtuse angle, simply do 180-(acute angle).

It's just a different way to approach it.

(edited 6 years ago)

Reply 8

Notnek

21

Original post by pondsteps

doesnt the one in bold contradict what u said before it?? :frown:

the formula always gives the acute angle between two vectors then how did it give us the obtuse angle in the top example?

If you had two vectors like in the diagram then

\displaystyle \frac{\mathbf{a}\cdot \mathbf{b}}{|\mathbf{a}||\mathbf {b}|}

would give you a negative value so

\theta

would be obtuse when you do inverse cos.

What I'm saying is that there's another formula that can be used that always gives you the acute angle. This works since it takes the modulus of the result before doing inverse cos.

Here is the standard formula:

\displaystyle \cos \theta = \frac{\mathbf{a}\cdot \mathbf{b}}{|\mathbf{a}||\mathbf {b}|}

And here is the formula that always gives you the acute angle:

\displaystyle \cos \theta = \left|\frac{\mathbf{a}\cdot \mathbf{b}}{|\mathbf{a}||\mathbf {b}|}\right|

You don't need to know this formula but it is mentioned in the Edexcel textbook for example. Instead you could just subtract from 180 if a question asks for the acute angle and you get an obtuse angle using the standard formula.

Reply 9

alevels2k17

OP

14

ok so this was my exact question... he was finding the angle between two intersecting lines and they were both in the same direction... he got 110.whatever.... so isnt that supposed to be the angle between the two lines! since the rule gives us the angle between two vecctors in the same direction right!! how come he did 180-110+ 61 and took that as the amgle???

Original post by pondsteps

ok so this was my exact question... he was finding the angle between two intersecting lines and they were both in the same direction... he got 110.whatever.... so isnt that supposed to be the angle between the two lines! since the rule gives us the angle between two vecctors in the same direction right!! how come he did 180-110+ 61 and took that as the amgle???

Can you please post the question?

Remember that it's hard to know if the angle will be obtuse or acute before drawing the diagram so this would be why his diagram looks like the angle is acute but it's actually obtuse.

Reply 11

alevels2k17

OP

14

Original post by notnek

Can you please post the question?

Remember that it's hard to know if the angle will be obtuse or acute before drawing the diagram so this would be why his diagram looks like the angle is acute but it's actually obtuse.

here it is

Original post by pondsteps

here it is

The question specifically asks for the acute angle between the lines. The situation is just like I showed with the blue/red angles. He has used the formula and ended up with an obtuse angle (blue) and so he has to subtract from 180 to give the acute (red) angle.

Reply 13

alevels2k17

OP

14

Original post by notnek

The question specifically asks for the acute angle between the lines. The situation is just like I showed with the blue/red angles. He has used the formula and ended up with an obtuse angle (blue) and so he has to subtract from 180 to give the acute (red) angle.

okay so the dot product rule is giving us the angle between the two lines in the same direction.. so in this case it gave us the obtuse meaning that is the angle.. why did he do 180-110 and out 60 as the angle if that isnt the one we got from the dot product!! - 110 is the angle between the lines going in the same direction (im asking cos we use this angle later in the question)

im so sorry i really suck at vectors :frown:

(edited 6 years ago)

Reply 14

Philip-flop

18

Original post by pondsteps

doesnt the one in bold contradict what u said before it?? :frown:

the formula always gives the acute angle between two vectors then how did it give us the obtuse angle in the top example?

The standard formula

\displaystyle \cos \theta = \frac{\mathbf{a}\cdot \mathbf{b}}{|\mathbf{a}||\mathbf {b}|}

will always give you the angle of which two vectors meet (intersect) and are going in the same direction. So it could either be the acute angle or the obtuse angle (depending on the vectors that you have).

Tbh you'll know whether your calculator has given you the obtuse angle as it'll be >90 degrees. So if you're trying to find the acute angle then it'll be... acute angle = 180 - obtuse angle

(edited 6 years ago)

Original post by Philip-flop

The standard formula

\displaystyle \cos \theta = \frac{\mathbf{a}\cdot \mathbf{b}}{|\mathbf{a}||\mathbf {b}|}

will always give you the angle of which two vectors are going in the same direction. So it could either be the acute angle or the obtuse angle (depending on the vectors that you have).

Tbh you'll know whether your calculator has given you the obtuse angle as it'll be >90 degrees. So if you're trying to find the acute angle then it'll be... acute angle = 180 - obtuse angle

Original post by pondsteps

okay so the dot product rule is giving us the angle between the two lines in the same direction.. so in this case it gave us the obtuse meaning that is the angle.. why did he do 180-110 and out 60 as the angle if that isnt the one we got from the dot product!! (im asking cos we use this angle later in the question)

im so sorry i really suck at vectors :frown:

Be careful when using the term "direction" here... With vectors, if two vectors have the same direction then they are parallel so there is no angle in between them.

@pondsteps Because again, there are two angles between two non-perpendicular lines which intersect, one obtuse and one acute. You do not know which one of these the formula

\displaystyle \cos \theta = \frac{\mathbf{a}\cdot \mathbf{b}}{|\mathbf{a}||\mathbf {b}|}

will give you until you calculate

{\bf a} \cdot {\bf b}

and determine its sign.

(edited 6 years ago)

Reply 16

Notnek

21

Original post by pondsteps

why did he do 180-110 and out 60 as the angle if that isnt the one we got from the dot product!! (im asking cos we use this angle later in the question)

As I showed above, there will generally be an obtuse and an acute angle between two vectors. The dot product formula will give you the angle between the two vectors pointing away from or towards each other. This could be acute or obtuse.

The question doesn't ask for the angle that you get using the dot product formula, it specificallly asks you for the acute angle. If you were to use the dot product formula for this question and get an acute angle then you wouldn't need to do anything. But since you end up with an obtuse (blue) angle, you need to subtract from 180 to get the (red) acute angle.

im so sorry i really suck at vectors :frown:

This is a common thing that confuses students so don't worry.

(edited 6 years ago)

Reply 17

alevels2k17

OP

14

Original post by RDKGames

Be careful when using the term "direction" here... With vectors, if two vectors have the same direction then they are parallel so there is no angle in between them.

@pondsteps Because again, there are two angles between two lines which intersect, one obtuse and one acute. You do not know which one of these the formula

\displaystyle \cos \theta = \frac{\mathbf{a}\cdot \mathbf{b}}{|\mathbf{a}||\mathbf {b}|}

will give you until you calculate

{\bf a} \cdot {\bf b}

and determine its sign.

ohhhhh omg really... so what does the sign of a.b tell us about the angle?

Reply 18

Philip-flop

18

Original post by RDKGames

Be careful when using the term "direction" here... With vectors, if two vectors have the same direction then they are parallel so there is no angle in between them.

@pondsteps Because again, there are two angles between two non-perpendicular lines which intersect, one obtuse and one acute. You do not know which one of these the formula