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

Question

BlackTom AI · Accepted Answer

We are asked to find the area of the parallelogram spanned by two vectors in R^3. The area of a parallelogram formed by vectors a and b is given by the magnitude of their cross product, |a × b|.

First, denote the vectors: a = (1, 1, 0) and b = (1, -1, 0).

Compute the cross product a × b. Using the determinant form:
 a × b = |i  j  k|
          |1  1  0|
          |1 -1  0|
This yields: i*(1*0 - 0*(-1)) - j*(1*0 - 0*1) + k*(1*(-1) - 1*1).
Simplifying each component:
 - i-component: 0 - 0 = 0
 - j-component: 0 - 0 = 0 (note the minus sign gives 0)
 - k-component: (1*(-1) - 1*1) = -1 - 1 = -2
So a × b = (0, 0, -2).

The area is the magnitude of this cross product: |a × b| = sqrt(0^2 + 0^2 + (-2)^2) = sqrt(4) = 2.

Therefore, the area of the parallelogram spanned by the given vectors is 2.

If you were verifying via an alternative route, you could also compute the parallelogram area as the product of the lengths of a and b times the sine of the angle between them, or by building a 2×2 matrix with the vector components in a plane and taking its determinant; in this case, both approaches yield the same result since the vectors lie in the xy-plane and the cross product effectively captures the perpendicular area contribution.

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

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

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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 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 (,,*).

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} ?

[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

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

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

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 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 (*,*,*).

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} ?

[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

希望你的学习变得更简单

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 (,,*).