Calculating QP, QR, and the Angle PQR

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In summary, the conversation discusses three points with position vectors p, q, and r, and their respective vectors QP and QR. It is shown that QP and QR are not perpendicular. The angle PQR is determined using the formula a.b=|QP|.|QR|cos(theta). The final answer for theta is approximately 0.8421613497 or 32.6º.
  • #1
lemon
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1. Three points P, Q and R have position vectors p, q and r respectively, where:
p=8i+11j, q=7i-5j and r=2i+4j.
Write down the vectors QP and QR and show that they are not perpendicular. Hence determine the angle PQR.




2. |QP|.|QR|cos(theta)




3. QP= QO+OP=i+16j
QR=QO+OR=-5i+9j
|QP|=root257
|QR|=root106

If the vectors are not perpendicular then a.b=0
a.b= QP=i+16j x QR=-5i+9j = (1)(-5)+(16)(9)=-5+144=139 - not perpendicular

a.b=|QP|.|QR|cos(theta)
cos(theta)=a.b/|QP|.|QR|=139/root257 x root106 = 32.6


Could anybody check to see how pathetic this attempt is, please?
 
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  • #2
Hi lemon! :smile:

Yes, that looks fine, except for the very last bit (the 32.6). :wink:
 
  • #3
errr. can't see it
thats the same figure i keep getting out.

?
 
  • #4
lemon said:
cos(theta)=a.b/|QP|.|QR|=139/root257 x root106 = 32.6

i don't know how you get that :redface:

139/√257√106 is approximately 139/16*10 < 1 :confused:
 
  • #5
yeah but that is cos(theta), right?
so to find theta i need to inverse cos - 32.6
 
  • #6
lemon said:
yeah but that is cos(theta), right?
so to find theta i need to inverse cos - 32.6

But it isn't 32.6! :cry:
 
  • #7
it is on my calculator. I get 0.8421613497
inverse cos = 32.63093847
 
  • #8
errr. can't see it
thats the same figure i keep getting out.

?
 
  • #9
lemon said:
it is on my calculator. I get 0.8421613497
inverse cos = 32.63093847

ohhh! you wrote cos(theta) = 32.6 …
lemon said:
cos(theta)=a.b/|QP|.|QR|=139/root257 x root106 = 32.6

yes, theta = 32.6º is fine. :smile:
 
  • #10
thanks tT
 

1. What is the formula for calculating QP and QR?

The formula for calculating QP and QR is the Pythagorean theorem: c^2 = a^2 + b^2, where c represents the hypotenuse (QR) and a and b represent the other two sides (QP and PR).

2. How do you find the angle PQR?

The angle PQR can be found using trigonometric functions, specifically the inverse tangent function. The formula for this is tan^-1(QP/QR) = PQR.

3. What do QP, QR, and PQR represent in a triangle?

QP and QR represent the lengths of the two sides connected to angle PQR, while PQR represents the measure of the angle itself in degrees or radians.

4. Can you use the Pythagorean theorem to solve for the angle PQR?

No, the Pythagorean theorem can only be used to calculate the lengths of sides in a right triangle. To find the measure of the angle PQR, trigonometric functions must be used.

5. Is it necessary to know the lengths of QP and QR to calculate the angle PQR?

Yes, the lengths of both QP and QR are needed to calculate the angle PQR using the inverse tangent function. Without these values, the angle cannot be accurately determined.

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