Show that the vectors 𝐚=⟨1,1,1⟩, 𝐛=⟨1,-1,0⟩, and 𝐜=⟨-1,-1,2⟩are mutually orthogonal, that is, ea

Name: Problem 12, Chapter 11: Calculus
Uploaded: 2021-06-15T15:35:58-08:00
Duration: 2 min 7 s
Channel: Rukhmani Jain
Description: Show that the vectors 𝐚=⟨1,1,1⟩, 𝐛=⟨1,-1,0⟩, and 𝐜=⟨-1,-1,2⟩are mutually orthogonal, that is, each pair of vectors is orthogonal.

step 1 In this question we are given with three vectors and the vector A is here 1, 1 and 1 and Vector

Question

Show that the vectors $\mathbf{a}=\langle 1,1,1\rangle, \mathbf{b}=\langle 1,-1,0\rangle,$ and $\mathbf{c}=\langle-1,-1,2\rangle$ are mutually orthogonal, that is, each pair of vectors is orthogonal.

   Show that the vectors $\mathbf{a}=\langle 1,1,1\rangle, \mathbf{b}=\langle 1,-1,0\rangle,$ and
$\mathbf{c}=\langle-1,-1,2\rangle$ are mutually orthogonal, that is, each pair of vectors is orthogonal.

Calculus

Dale Varberg, Edwin… 9th Edition

Chapter 11, Problem 12

Instant Answer

Solved by Expert Rukhmani Jain

06/15/2021

Step 1

The dot product of two vectors is calculated by multiplying their corresponding components and adding the results. So, we have: \[\mathbf{a} \cdot \mathbf{b} =\langle 1,1,1\rangle \cdot\langle 1,-1,0\rangle =1(1)+1(-1)+1(0) =0.\] Show more…

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Show that the vectors $\mathbf{a}=\langle 1,1,1\rangle, \mathbf{b}=\langle 1,-1,0\rangle,$ and $\mathbf{c}=\langle-1,-1,2\rangle$ are mutually orthogonal, that is, each pair of vectors is orthogonal.

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Key Concepts

Dot Product

The dot product, also known as the inner product, is a mathematical operation that multiplies corresponding components of two vectors and sums the results. This operation is crucial for determining angles between vectors, and a zero dot product is a key indicator of orthogonality.

Mutually Orthogonal

A set of vectors is said to be mutually orthogonal if every pair of distinct vectors within the set has a dot product equal to zero. This property ensures that each vector in the set is perpendicular to every other vector, which is important in constructing orthogonal bases in vector spaces.

Orthogonality

Orthogonality refers to the condition where two vectors have a dot product of zero, indicating that they are perpendicular to each other. In Euclidean space, this means that there is no projection of one vector along the direction of the other, making them independent in terms of direction.

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