Find the unit vectors perpendicular to the given vectors

In summary: Which means that the equation has been solved for ##a##.No, the following would not be another equation in the variables a, b and c due to the fact that the unknown vector is a unit vector. If yes, then we end up with 3 equations for the three variables, which now can be solved for a, b and c.$$a^2 + b^2 + c^2= 1$$We could find ##a## in terms of ##b## from first equation. Then from second equation, substituting ##a## in...$$\vec{a} = \frac{1}{b
  • #1
brochesspro
155
22
Homework Statement
Given below.
Relevant Equations
Given below.
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  • #2
Check the equation beneath ##-a+4b-5c=0##. :)

(can you solve it faster another way? what are the properties of the cross product?)
 
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  • #3
Yeah I’m going to have to echo the comment about using cross products.

Take the cross product of the two vectors and find its unit vector. Then negate it and you have your two answers.
 
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  • #4
ergospherical said:
(can you solve it faster another way? what are the properties of the cross product?)
Do you mean dot product?
 
  • #5
brochesspro said:
Do you mean dot product?
vector1-1.png
 
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  • #6
brochesspro said:
Do you mean dot product?
I think they meant the cross product!
 
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  • #7
PhDeezNutz said:
Yeah I’m going to have to echo the comment about using cross products.

Take the cross product of the two vectors and find its unit vector. Then negate it and you have your two answers.
You gave me the answer too soon. It would have been better if you would have slowly led me to that conclusion.
 
  • #8
PeroK said:
I think they meant the cross product!
I was only focusing on the perpendicular part and the fact that perpendicular vectors have dot product 0. And so, I completely forgot about cross products.
 
  • #9
But still, thank you for the responses. Now I can solve this question on my own.
 
  • #10
brochesspro said:
You gave me the answer too soon. It would have been better if you would have slowly led me to that conclusion.

Fair enough. Let’s back track. Can you prove that. Let’s justify this approach.

##(\vec{a} \times \vec{b}) \cdot \vec{b} = (\vec{a} \times \vec{b}) \cdot \vec{a} = 0##
Cross-product-formula.png
 
  • #11
brochesspro said:
I was only focusing on the perpendicular part and the fact that perpendicular vectors have dot product 0. And so, I completely forgot about cross products.
That's a good point. You could look generate simultanenous equations by looking for a vector whose dot product with both those vectors is zero. Using the cross product could be quicker.
 
  • #12
PhDeezNutz said:
Fair enough. Let’s back track. Can you prove that. Let’s justify this approach.

##(\vec{a} \times \vec{b}) \cdot \vec{b} = (\vec{a} \times \vec{b}) \cdot \vec{a} = 0##View attachment 295004
No need for it now, I already got the answer. But, thank you.
 
  • #13
PeroK said:
That's a good point. You could look generate simultanenous equations by looking for a vector whose dot product with both those vectors is zero. Using the cross product could be quicker.
I tried to do that itself. Could you once go through the attachments and guide me to what I have done wrong?
 
  • #14
brochesspro said:
I tried to do that itself. Could you once go through the attachments and guide me to what I have done wrong?
Never mind, I got the answer. Again, thank you for all your responses.
 
  • #15
brochesspro said:
I tried to do that itself. Could you once go through the attachments and guide me to what I have done wrong?
You got as far as $$2a -3b = 0$$$$-a + 4b - 5c =0$$And then things seemed to go a bit wrong. I just multiplied the second equation by ##2## and then added them.

Note that you are always going to have a free parameter here. In this case ##c## can be anything.
 
  • #16
PS I like to write vectors as ##(2, -3, 0), (-1, 4, -5), (a, b, c)##. Has this gone out of fashion? Anyway, it seems very simple to write down the dot products in this way.
 
  • #17
PeroK said:
PS I like to write vectors as ##(2, -3, 0), (-1, 4, -5), (a, b, c)##. Has this gone out of fashion? Anyway, it seems very simple to write down the dot products in this way.
I am not sure about it. But, many people write it that way, I think it is used for points rather than vectors, but I may be wrong.
PeroK said:
You got as far as $$2a -3b = 0$$$$-a + 4b - 5c =0$$And then things seemed to go a bit wrong. I just multiplied the second equation by ##2## and then added them.

Note that you are always going to have a free parameter here. In this case ##c## can be anything.
Yes, I noticed that I forgot to multiply the 4 by 2 and things went downhill from there. But, I can now do this both ways, the dot product way, and the cross product way. Goodbye for now.
 
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  • #18
PeroK said:
You got as far as $$2a -3b = 0$$$$-a + 4b - 5c =0$$And then things seemed to go a bit wrong. I just multiplied the second equation by ##2## and then added them.

Note that you are always going to have a free parameter here. In this case ##c## can be anything.

Wouldn't the following be another equation in the variables a, b and c due to the fact that the unknown vector is a unit vector? If yes, then we end up with 3 equations for the three variables, which now can be solved for a, b and c.
$$a^2 + b^2 + c^2= 1$$We could find ##a## in terms of ##b## from first equation. Then from second equation, substituting ##a## in terms of ##b##, we will get ##c## in terms of ##b##. Now, we substitute in above equation the variables ##a## and ##c##, and solve for ##b##. Knowing ##b## means we can determine ##a## and ##c##.
 
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  • #19
vcsharp2003 said:
Wouldn't the following be another equation in the variables a, b and c due to the fact that the unknown vector is a unit vector? If yes, then we end up with 3 equations for the three variables, which now can be solved for a, b and c.
$$a^2 + b^2 + c^2= 1$$We could find ##a## in terms of ##b## from first equation. Then from second equation, substituting ##a## in terms of ##b##, we will get ##c## in terms of ##b##. Now, we substitute in above equation the variables ##a## and ##c##, and solve for ##b##. Knowing ##b## means we can determine ##a## and ##c##.
That is what I did, but with ##c##.
 
  • #20
brochesspro said:
That is what I did, but with ##c##.
That should give you two different unit vectors.
 
  • #21
Yes, I got them.
 

Related to Find the unit vectors perpendicular to the given vectors

1. What is a unit vector?

A unit vector is a vector with a magnitude of 1. It represents the direction of a given vector without any influence from its magnitude.

2. How do you find the unit vector perpendicular to a given vector?

To find the unit vector perpendicular to a given vector, you can use the cross product or dot product method. Both methods involve mathematical calculations using the given vector's components.

3. What is the importance of finding unit vectors perpendicular to a given vector?

Unit vectors perpendicular to a given vector help in determining the direction of that vector in a three-dimensional space. They are also useful in physics and engineering applications, such as calculating forces and determining the direction of motion.

4. Can there be more than one unit vector perpendicular to a given vector?

Yes, there can be infinitely many unit vectors perpendicular to a given vector. This is because there are infinite directions in a three-dimensional space that can be perpendicular to a given vector.

5. What is the difference between a unit vector and a perpendicular vector?

A unit vector has a magnitude of 1 and represents the direction of a given vector, while a perpendicular vector is a vector that forms a 90-degree angle with the given vector. A unit vector can also be perpendicular to a given vector, but not all perpendicular vectors are unit vectors.

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