Finding 'p' for Perpendicular Vectors: A+B=0?

In summary, to find the constant 'p' such that vectors a and b are perpendicular, we use the dot product and set it equal to zero. Solving algebraically, we get p = 1/3 and p = 1 as the solutions. This means that the angle between the two vectors is 90 degrees. The vector cross product can also be used to find a vector perpendicular to both a and b.
  • #1
escobar147
31
0
for the following two vectors find the constant 'p' such that the vectors a & b are perpendicular:

a = i + 2pj +3pk

b = i - 2j + pk

the answer is: p = 1 & p = 1/3, but how is this calculated? any help would be massively appreciated

i understand that the angle between the vectors, when perpendicular would be 90 degrees and apparently the dot product should be equal to zero, however the answer would suggest otherwise?
 
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  • #2
if two vectors are perpendicular to each other then that means the angle between them is 90 degrees.

what vector operation gives you the angle between them? Use it to determine p.

Check your math especially the signage. It worked for me.
 
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  • #3
jedishrfu said:
if two vectors are perpendicular to each other then that means the angle between them is 90 degrees.

what vector operation gives you the angle between them? Use it to determine p.

Check your math especially the signage. It worked for me.

ab=sin90??
 
  • #4
The "operation" jdishrfu is talking about is the dot product. Two vectors are perpendicular if and only if their dot product is 0. However, "p= 1/3" is NOT a correct solution.
 
  • #5
HallsofIvy said:
The "operation" jdishrfu is talking about is the dot product. Two vectors are perpendicular if and only if their dot product is 0. However, "p= 1/3" is NOT a correct solution.

maybe I'm wrong but plugging p=1/3 into the original vectors and dotting them together I get zero.

a = i + 2/3j + k

b = i - 2j + 1/3k

a.b = 1 -4/3 + 1/3 = 4/3 - 4/3 = 0 --> cos 90 = 0
 
  • #6
escobar147 said:
ab=sin90??

I think you're referring to the vector cross product to find a vector that is perpendicular to both vectors.
 
  • #7
Sorry, you are right. For some reason I missed the "p" in the j component of the first vector.
 
  • #8
sorry it seems i was making it a lot harder than it needed to be, it's just a case of solving algebraically:

ab = (1, 2p, 3p) (1, -2, p)
= (1)(1) + (2p)(-2) + (3p)(p)
= 3p^2 - 4p + 1.

then, putting equations equal to zero and solving for p:
3p^2 - 4p + 1 = 0 = p = 1/3 and p = 1.
 
  • #9
thanks for your help
 

1. How do I find the value of 'p' for perpendicular vectors A and B?

To find the value of 'p' for perpendicular vectors A and B, you can use the formula p = -(A dot B)/(A dot A), where 'dot' represents the dot product of the two vectors. This formula is derived from the fact that for perpendicular vectors, the dot product is equal to 0.

2. What is the significance of finding 'p' for perpendicular vectors A and B?

Finding 'p' for perpendicular vectors A and B is important because it allows us to determine the length of the projection of A onto B, or vice versa. This can be useful in various applications, such as calculating the forces acting on an object in physics or determining the direction of motion in navigation systems.

3. Can I use this formula for any two perpendicular vectors?

Yes, this formula can be used for any two perpendicular vectors. It is a general formula that applies to all cases of perpendicular vectors, regardless of their magnitude or direction.

4. What if the vectors A and B are not perpendicular?

If the vectors A and B are not perpendicular, the value of 'p' will not be meaningful. In fact, the dot product of the two vectors will not be equal to 0, which is the condition for perpendicular vectors. In this case, a different approach or formula will need to be used to find the value of 'p'.

5. Are there any other methods to find 'p' for perpendicular vectors A and B?

Yes, there are other methods to find the value of 'p' for perpendicular vectors A and B, such as using the Pythagorean theorem or the cross product of the two vectors. However, the formula p = -(A dot B)/(A dot A) is the most commonly used and efficient method for finding 'p'.

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