Answer :
For the triangle with the vertices:
QP is the vector from point Q to point P, and it is (-1, -3, 2).
QR is the vector from point Q to point R, and it is (4, -2, -1).
The dot product QP ⋅ QR is expressed as cos(θ) × (√14 × √21),
How to find vectors?
To calculate QP:
QP is the vector that goes from point Q to point P. To find QP, subtract the coordinates of Q from the coordinates of P.
QP = P - Q
= (-1, -4, -4) - (0, -1, -6)
= (-1, -4, -4) - (0, -1, -6)
= (-1 - 0, -4 + 1, -4 + 6)
= (-1, -3, 2)
To calculate QR:
QR is the vector that goes from point Q to point R. To find QR, subtract the coordinates of Q from the coordinates of R.
QR = R - Q
= (4, -3, -7) - (0, -1, -6)
= (4, -3, -7) - (0, -1, -6)
= (4 - 0, -3 + 1, -7 + 6)
= (4, -2, -1)
To calculate the dot product QP ⋅ QR:
The dot product of two vectors A and B is given by A ⋅ B = |A| × |B| × cos(θ), where |A| and |B| are the magnitudes of the vectors, and θ is the angle between them. Since you want the dot product of QP and QR, calculate it as follows:
QP ⋅ QR = |QP| × |QR| × cos(θ)
First, find the magnitudes of QP and QR:
|QP| = √((-1)² + (-3)² + 2²)
= √(1 + 9 + 4)
= √14
|QR| = √(4² + (-2)² + (-1)²)
= √(16 + 4 + 1)
= √21
Now, find the cosine of the angle θ between QP and QR. Use the dot product formula rearranged:
cos(θ) = (QP ⋅ QR) / (|QP| × |QR|)
Plug in the values:
cos(θ) = (QP ⋅ QR) / (√14 × √21)
Now, solve for QP ⋅ QR:
QP ⋅ QR = cos(θ) × (√14 × √21)
So, QP = (-1, -3, 2), QR = (4, -2, -1), and QP ⋅ QR = cos(θ) × (√14 × √21).
Find out more on vectors here: https://brainly.com/question/28028700
#SPJ1
Final answer:
To find QP and QR, subtract the coordinates of Q from P and R respectively. The dot product of QP and QR is 0.
Explanation:
The triangle with vertices P(−1,−4,−4),Q(0,−1,−6), and R(4,−3,−7).
To find the vector QP, we subtract the coordinates of point Q from the coordinates of point P.
QP = P - Q = (-1, -4, -4) - (0, -1, -6) = (-1-0, -4+1, -4-(-6)) = (-1, -3, 2).
To find the vector QR, we subtract the coordinates of point Q from the coordinates of point R.
QR = R - Q = (4, -3, -7) - (0, -1, -6) = (4-0, -3+1, -7-(-6)) = (4, -2, -1).
To find the dot product of QP and QR, we multiply the corresponding components and sum them up.
QP ⋅ QR = (-1)(4) + (-3)(-2) + (2)(-1) = -4 + 6 - 2 = 0.
Learn more about Vectors here:
https://brainly.com/question/33923402
#SPJ1
