Finding Perpendicular Vectors with Sum [6,8]

In summary: Sure thing. Good luck!In summary, the conversation discusses the need to find two perpendicular vectors, u and v, where u is twice the magnitude of v and their sum vector is [6, 8]. The summary also mentions the use of equations involving |u|^2 and |v|^2 to solve for the unknown variables a, b, c, and d in order to solve the problem.
  • #1
PiRsq
112
0
Find 2 vectors u and v such that they are perpendicular one of the vector is twice the magnitude of the other. And the sum vector of u and v is [6,8]

I did:

Let u=[a,b]
Let v=[c,d]
Let |u|=2|v

u.v=ac+bd=0

|u+v|=|u|^2 + |v|^2

But |u|=2|v|

|u+v|=5|v|^2

5|v|^2=100

|v|^2=20

Im stuck after this
 
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  • #2
You forgot to write ^2 on |u+v|

Anyways, can you write |v|^2 in terms of a, b, c, and d? Do you know what |u|^2 is? Have you used the fact that u + v = [6, 8] yet?
 
  • #3
So

|u|^2+|v|^2=|u+v|^2

But since |u|^2=4|v|^2

5|v|^2=36+64
|v|^2=20

now where can I go?
 
  • #4
Can you write |v|^2 in terms of a, b, c, and d? Do you know what |u|^2 is? Have you used the fact that u + v = [6, 8] yet?
 
  • #5
|v|^2 = c^2+d^2

|v|^2 = 1/2 (a^2 + b^2)


1/2(a^2+ b^2)= c^2 + d^2

Is that what you mean?
 
  • #6
Can you think of anything better you can do with those first two equatinos?
 
  • #7
Since |v|^2=2|u|^2,

|v|^2 = c^2+d^2

|u|^2 = 1/2 (c^2 + d^2)


|u|^2 + |v|^2 = 100

3/2 (c^2 + d^2) = 100

I don't know where I am headed
 
  • #8
what formulas do you have involving |v|^2?
 
  • #9
Projection of u on v

u on v = [u.v/|v|^2] |v|
 
  • #10
I guess I should have asked this first off...

You know that your goal is to find 4 equations involving only a, b, c, d which you can solve, right?
 
  • #11
4 equations?? I didnt know that...I've been focusing on finding a and b
 
  • #12
Well, you have 4 unknowns; a, b, c, and d. In general, you need 4 equations to solve for all 4 of them.

Sometimes you can get lucky and you can find two equations that involve only a and b, but in general that won't happen (and I'm pretty sure it doesn't here)...


You've already found one good equation:

ac + bd = 0


You just need 3 more! You can get 2 more equations out of what you've told me about |v|^2...


Oh, BTW, if |u| = 2|v|, then |u|^2 = 4|v|^2
 
  • #13
Ill try this and post a little later, thanks man
 

1. How do I find perpendicular vectors with a sum of [6,8]?

To find perpendicular vectors with a sum of [6,8], you can use the formula a = [a1, a2] and b = [b1, b2], where a and b are perpendicular vectors. The dot product of these two vectors will be zero, so you can set up the equation a1b1 + a2b2 = 0. From there, you can solve for either a1 or a2, and then use that value to find the other component of the vector.

2. What is the significance of [6,8] in finding perpendicular vectors?

[6,8] represents the sum of the two perpendicular vectors that you are trying to find. This means that when you add the components of the two vectors together, you will get [6,8]. This is important because it helps you set up the equation to solve for the components of the vectors.

3. Can perpendicular vectors have different magnitudes?

Yes, perpendicular vectors can have different magnitudes. As long as the dot product of the two vectors is zero, they are considered perpendicular. This means that the angle between the two vectors is 90 degrees, but the lengths of the vectors can be different.

4. How can I visualize perpendicular vectors with a sum of [6,8]?

You can visualize perpendicular vectors with a sum of [6,8] by graphing them on a coordinate plane. Start by plotting the two components of each vector as points on the plane. Then, draw a line from the origin to each point. If the two lines are perpendicular, then the vectors are perpendicular. You can also use a vector diagram to visualize the relationship between the two vectors and their sum.

5. Are there any real-world applications of finding perpendicular vectors with a sum of [6,8]?

Yes, there are many real-world applications of finding perpendicular vectors with a sum of [6,8]. For example, this concept is used in physics to calculate the torque on an object. The perpendicular vectors represent the force and the distance from the pivot point, and their sum represents the torque. This concept is also used in engineering, navigation, and computer graphics.

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