Perpendicular Vector with given magnitude

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To find a vector perpendicular to A = 8i - 5j with a magnitude of 14, the dot product method reveals potential perpendicular vectors such as B = -5i - 8j or B = 5i + 8j. However, these vectors need to be scaled to achieve the desired magnitude of 14. The magnitude formula √(a² + b²) must equal 14, which requires multiplying the perpendicular vectors by appropriate scalar values. The discussion emphasizes that the scaling factors do not have to be integers, allowing for various solutions. Understanding how to apply these multiples is crucial for solving the problem effectively.
NEUhusky
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Homework Statement



Given a vector A = 8i − 5j, find the vectors in the xy plane that are perpendicular to A and have a magnitude of 14.

The Attempt at a Solution



I can find the vector perpendicular to A quite easily. I'm having trouble finding the perpendicular line when it has the given magnitude of 14. I have a feeling it's going to be an easy solution but something isn't clicking in my brain at the moment. Any help would be greatly appreciated. Thanks!
 
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Welcome to PF!

Hi NEUhusky! Welcome to PF! :wink:

The magnitude of ai + bj is √(a2 + b2) …

show us what you get. :smile:​
 


tiny-tim said:
Hi NEUhusky! Welcome to PF! :wink:

The magnitude of ai + bj is √(a2 + b2) …

show us what you get. :smile:​

Thanks! Glad I found this place! :smile:

Hm, well the magnitude of the given vector A is √(8^2+5^2) which is approximately 9.43.

Using the dot product I know that the perpendicular lines are either B = -5i-8j or 5i+8j because the dot product equals zero. Unfortunately the magnitude is still the same as A when it needs to be 14. Still kinda lost.
 
ah!

no, 5i + 8j is a point on the perpendicular line …

multiply it by anything, and you'll get more points on the perpendicular line …

choose the correct multiples to get 14. :wink:
 
tiny-tim said:
ah!

no, 5i + 8j is a point on the perpendicular line …

multiply it by anything, and you'll get more points on the perpendicular line …

choose the correct multiples to get 14. :wink:

Ugh, still a bit confused. Multiples to get 14 would be either 2 & 7 or 1 & 14. I'm not sure how that applies. Doesn't the magnitude ( √(a2 + b2) ) have to equal 14?
 
Hi NEUhusky! :smile:

(just got up :zzz: …)
NEUhusky said:
Ugh, still a bit confused. Multiples to get 14 would be either 2 & 7 or 1 & 14. I'm not sure how that applies. Doesn't the magnitude ( √(a2 + b2) ) have to equal 14?

(try using the X2 icon just above the Reply box :wink:)

Yes, but it doesn't have to be an integer! :wink:
 

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