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Chapter 7: Problem 2

What is the dot product of two orthogonal vectors?

Short Answer

Expert verified
The dot product of two orthogonal vectors is 0.

Step by step solution

01

Definition of Orthogonal Vectors

Two vectors are considered orthogonal or perpendicular to each other if the angle between them is 90 degrees. This property can be described mathematically by the dot product, where the dot product of two vectors \( \vec{a} \) and \( \vec{b} \) is defined as \( \vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos(\theta) \), where \( \theta \) is the angle between the vectors.
02

Dot Product of Orthogonal Vectors

For orthogonal vectors, the angle between the vectors is 90 degrees. The cosine of 90 degrees is 0. Therefore, the dot product between two orthogonal vectors equals 0 because \( \vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos(90^{\circ}) = 0 \), regardless of the length of the vectors.

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Most popular questions from this chapter

Find the magnitude and direction angle of the vector v.$$\mathbf{v}=-7 \mathbf{i}-6 \mathbf{j}$$

Find the component form of \(v\) and sketch the specified vector operations geometrically, where \(\mathbf{u}=2 \mathbf{i}-\mathbf{j}\) and \(\mathbf{w}=\mathbf{i}+2 \mathbf{j}\). $$\mathbf{v}=-\mathbf{u}+\mathbf{w}$$

Sketch the graph of all complex numbers \(z\) satisfying the given condition. $$\theta=\frac{\pi}{4}$$

Find the projection of \(\mathbf{u}\) onto \(\mathbf{v}\). Then write \(\mathbf{u}\) as the sum of two orthogonal vectors, one of which is \({\mathrm{proj}_v}\mathrm{u}.\) $$\begin{aligned} &\mathbf{u}=\langle 0,3\rangle\\\ &\mathbf{v}=\langle 2,15\rangle \end{aligned}$$

Determine whether \(\mathbf{u}\) and \(\mathbf{v}\) are orthogonal. $$\begin{aligned} &\mathbf{u}=\langle 10,-6\rangle\\\ &\mathbf{v}=\langle 9,15\rangle \end{aligned}$$
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