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Chapter 6: Problem 82

If \(\mathbf{v}=-2 \mathbf{i}+5 \mathbf{j},\) find a vector orthogonal to \(\mathbf{v}\)

Short Answer

Expert verified
The vector orthogonal to vector \(\mathbf{v}=-2 \mathbf{i}+5 \mathbf{j}\) is \(\mathbf{u}=\mathbf{i}+(2/5)\mathbf{j}\).

Step by step solution

01

Understand the Orthogonal Property

Two vectors are orthogonal if and only if the dot product of the vectors equals zero. In our case, let's assume a vector \(\mathbf{u}=x\mathbf{i}+y\mathbf{j}\), we can calculate the dot product as: \(\mathbf{u} \cdot \mathbf{v} = x*(-2) + y*5\). We want this to equal zero since we are looking for an orthogonal vector.
02

Finding the vector components

To make computation easier, we can decide a value for one of the components of the new vector. For example we can take \(x=1\). We substitute \(x\) value in the dot product equation \(-2*1 + 5*y = 0\).
03

Solve for y

To find the y component of the vector, solve the equation from the previous step for y: \(-2 + 5*y = 0 \Longrightarrow y = 2/5\).
04

Verify the Solution

Verify the answer by computing the dot product of \(\mathbf{v}=-2 \mathbf{i}+5 \mathbf{j}\) and the orthogonal vector \(\mathbf{u}=\mathbf{i}+(2/5)\mathbf{j}\). The result should be zero if the vectors are truly orthogonal.

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