High School

Consider the triangle with vertices \( P(-1, -4, -4) \), \( Q(0, -1, -6) \), and \( R(4, -3, -7) \).

Determine the following vectors:
\[ QP = \]
\[ QR = \]

Find:
\[ QP \]
\[ QR \]
\[ QP \cdot QR \]

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).

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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:

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