Let [math: u]{\bf u} and [math: v]{\bf v} be vectors in R3. What is the value of [math: u×v⋅v]{\bf u} imes {\bf v} \cdot {\bf v} ?

Question

Let [math: u]{\bf u} and [math: v]{\bf v} be vectors in R3. What is the value of [math: u×v⋅v]{\bf u} 	imes {\bf v} \cdot {\bf v} ?

BlackTom AI · Accepted Answer

The problem asks for the value of (u × v) · v for vectors u, v in R^3.

One key fact: the cross product u × v is perpendicular to both u and v. Therefore, the dot product (u × v) · v must be zero because the result of u × v is orthogonal to v.

Another way to see this is to use the scalar triple product identity: (u × v) · v = u · (v × v). Since v × v = 0 (the cross product of any vector with itself is the zero vector), we have u · 0 = 0. Thus the value is 0.

A third perspective is to view the expression as a determinant with two identical columns when written in component form, which evaluates to zero as well.

In summary, all common approaches show that (u × v) · v equals 0, due to perpendicularity and the zero result of v × v.

MTH1030 -1035 - S1 2025 MTH1030/5 MTH1030/5 Week 1 lesson quiz: Vectors, lines and planes

Let [math: u]{\bf u} and [math: v]{\bf v} be vectors in R3. What is the value of [math: u×v⋅v]{\bf u} \times {\bf v} \cdot {\bf v} ?

查看解析

登录即可查看完整答案

类似问题

A vector that is orthogonal (perpendicular) to both vectors 𝑎 _ = ⟨ − 2 , − 5 , 1 ⟩ and 𝑏 _ = ⟨ 1 , 3 , − 2 ⟩ is: Hint: Vectors 𝑣 _ and 𝑤 _ are orthogonal if and only if 𝑣 _ ⋅ 𝑤 _ = 0 .

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?

Determine the result of the cross product of u=(1,0,0) and v=(0,1,0).(K:1)

What is the area of the parallelogram spanned by the vectors (1,1,0) and (1,-1,0)?

What is the cross-product of the two vectors (1,2,3) and (4, 5, 6)? Make sure you do this one by hand. Input your answer in the form (,,*).

[1 1 2] x [1 2 3] =nb: present result as the sum of the components. Ex.: [A B C]=A+B+C

[1 1 2] x [4 0 3] =nb: present result as the sum of the components. Ex.: [A B C]=A+B+C

更多留学生实用工具

考试浏览器助手

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