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

[escobar147]

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?

 

[jedishrfu]

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.

 

[escobar147]

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??

 

[HallsofIvy]

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.

 

[jedishrfu]

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

 

[jedishrfu]

ab=sin90??

I think you're referring to the vector cross product to find a vector that is perpendicular to both vectors.

 

[HallsofIvy]

Sorry, you are right. For some reason I missed the "p" in the j component of the first vector.

 

[escobar147]

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.

 

[escobar147]

thanks for your help