Let $P$, $Q$ and $R$ be the points $$P = \left (-3, -1, 2\right ), Q = \left (-1, 2, -2\right ), R = \left (2, 3, 0\right ).$$

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 $$P = \left (-3, -1, 2\right ), Q = \left (-1, 2, -2\right ), R = \left (2, 3, 0\right ).$$

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

The problem asks for a unit vector orthogonal to the plane through P, Q, and R with a positive first component, and then to report its third component.

First, compute the vectors along the plane from P to Q and from P to R:
- PQ = Q - P = (-1 - (-3), 2 - (-1), -2 - 2) = (2, 3, -4)
- PR = R - P = (2 - (-3), 3 - (-1), 0 - 2) = (5, 4, -2)

Next, find a normal vector to the plane by taking the cross product n = PQ × PR:
- n_x = 3*(-2) - (-4)*4 = -6 - (-16) = 10
- n_y = -(2*(-2) - (-4)*5) = -(-4 - (-20)) = -(16) = -16
- n_z = 2*4 - 3*5 = 8 - 15 = -7
So a normal vector is n = (10, -16, -7).

To obtain a unit vector, divide by its magnitude. The magnitude is ||n|| = sqrt(10^2 + (-16)^2 + (-7)^2) = sqrt(100 + 256 + 49) = sqrt(405) = 9*sqrt(5).

Thus, a unit normal with a positive first component is u = n / ||n|| = (10, -16, -7) / (9*sqrt(5)).

Reading off the third component of this unit vector gives u_z = -7 / (9*sqrt(5)).

Therefore, the third component is -7/(9√5).

Let \(P\), \(Q\) and \(R\) be the points \[P = \left (-3, -1, 2\right ), Q = \left (-1, 2, -2\right ), R = \left (2, 3, 0\right ).\] Find a unit vector orthogonal to the plane through \(P\), \(Q\) and \(R\) that has a positive 1st component. What is its 3rd component?单项选择题

类似问题

What was the world's first widely adopted biodiversity policy?

The aim of the Paris Agreement is to ensure that the earth’s preindustrial temperature is not exceeded by more than º Celsius.

The IPCC's mission is to Blank ______.

Why have most international agreements for environmental protection been based on the honor system?

As was exhibited with the Montreal Protocol and the encouragement of reduced CFC production, trade can be an effective tool to compel compliance with international treaties.

The Stockholm Declaration and UNEP were the result of the Blank ______ first conference on the Human Environment in 1972.

In addition to the direct mission of the Convention on International Trade in Endangered Species (CITES), why was CITES so important?

更多留学生实用工具

智能学习助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单