Let $P$, $Q$ and $R$ be the points $$P = \left (-1, 2, -2\right ), Q = \left (-3, 3, -3\right ), R = \left (3, -3, 1\right ).$$

Find a unit vector orthogonal to the plane through $P$, $Q$ and $R$ that has a positive 1st component.

What is its 3rd component?

Question

Let $P$, $Q$ and $R$ be the points $$P = \left (-1, 2, -2\right ), Q = \left (-3, 3, -3\right ), R = \left (3, -3, 1\right ).$$

Find a unit vector orthogonal to the plane through $P$, $Q$ and $R$ that has a positive 1st component.

What is its 3rd component?

BlackTom AI · Accepted Answer

We are given three points P, Q, R and asked to find a unit vector orthogonal to the plane through them that has a positive first component, then report its third component.

First, compute two direction vectors in the plane: Q - P and R - P. From P = (-1, 2, -2), Q = (-3, 3, -3), R = (3, -3, 1):
- Q - P = (-3 - (-1), 3 - 2, -3 - (-2)) = (-2, 1, -1).
- R - P = (3 - (-1), -3 - 2, 1 - (-2)) = (4, -5, 3).

Next, take the cross product of these two vectors to get a normal to the plane: n = (Q - P) × (R - P).
Compute: n = |i  j  k; -2 1 -1; 4 -5 3| = (-2, 2, 6).
This can be simplified by factoring 2: n ∝ (-1, 1, 3).

A normal vector to the plane can therefore be taken as n0 = (-1, 1, 3). A unit vector in the direction of n0 is u0 = n0 / ||n0||, where ||n0|| = sqrt((-1)^2 + 1^2 + 3^2) = sqrt(11). Thus u0 = (-1, 1, 3) / sqrt(11).

However, the problem requires a unit vector orthogonal to the plane that has a positive first component. The vector with positive first component is the opposite direction: u = -u0 = (1, -1, -3) / sqrt(11).

Therefore, the third component of the required unit vector is (-3) / sqrt(11).

All steps confirm that whether you align with the positive first component, the third component of the unit normal remains -3/√11, matching the given option.

MATH1061/1002/1021 (ND) MATH1061 Canvas Quiz 5

Let \(P\), \(Q\) and \(R\) be the points \[P = \left (-1, 2, -2\right ), Q = \left (-3, 3, -3\right ), R = \left (3, -3, 1\right ).\] Find a unit vector orthogonal to the plane through \(P\), \(Q\) and \(R\) that has a positive 1st component. What is its 3rd component?

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